











%. 









^ 






V 
V 






'.: 



^ ^ 









oo 












tf*« 



^ 






H -7* 















«* 



PRACTICAL TREATISE 



ON 



GEARING 






oJ^Ls s JUQstS^ 



• o 



1 4* 3 I 



3 



SIXTH EDITION. 



PROVIDENCE, R. I. 

BROWN & SHARPE MANUFACTURING COMPANY. 



1900. 



TWO COPIES RECEIVED, 

Library of C8Bgr§«% 
(Jfflee of tb® 

APR 1 1900 

tegltter of Copyrights, 

. B3-7 

)C[00 



61040 



Entered according to Act of Congress, in the year lyuO by 

BROWN & SHARPE MFG. CO., 

In the Office of the Librarian of Congress at Washington. 

Registered at Stationers' Hall, London, Eng. 

AH rights reserved. 




O0-2V9? P& 



SECOND GOPY, 



PREFACE. 



This Book is made for men in practical life ; for those that 
would like to know how to construct gear wheels, but whose 
duties do not afford them sufficient leisure to acquire a technical 
knowledge of the subject. 



<\\»\1 

Af^.1^,1^ 



CONTENTS 



PAKT I . 
Chapter I. 



PAGE. 

Pitch Circle — Pitch — Tooth — Space — Addendum or Face — 

Flank — Clearance 1 

Chapter II. 

Classification — Sizing Blanks and Tooth Parts from Circular 

Pitch — Ce n ter D istan ce 5 

Chapter III. 
Single Curve Gears of 30 Teeth and more 9 

Chapter IV. 

Rack to Mesh with Single Curve Gears having 30 Teeth and 

more 12 

Chapter V. 

Diametral Pitch — Sizing Blanks and Teeth — Distance be- 
tween the Centers of Wheels 16 

Chapter VI. 

Single-Curve Gears, having Less than 30 Teeth — Gears and 

Racks to Mesh with Gears having Less than 30 Teeth... 20 

Chapter VII. 
Double-Curve Teeth— Gear of 15 Teeth— Rack 25 

Chapter VIII. 

Double-Curve Gears, having More and Less than 15 Teeth 

— Annnlar Gears 30 



VI CONTENTS. 

Chapter IX. 

PAGK. 

Bevel Gear Blanks 3-t 

Chapter X. 
Bevel Gears — Form and Size of Teeth — Cutting Teeth 41 

Chapter XI. 
Worm Wheels — Sizing Blanks of 32 Teeth and more G3 

Chapter XII. 

Sizing Gears when the Distance between the Centers and the 
Ratio of Speeds are fixed — General Remarks — Width of 
Face of Spur Gears — Speed of Gear Cutters — Table of 
Tooth Parts 79 



PART II. 



Chapter I. 
Tangent of Arc and Angle 8? 

Chapter II. 

Sine, Cosine and Secant — Some of their Applications in 

Machine Construction 93 

Chapter III. 

Application of Circular Functions — Whole Diameter of Bevel 

Gear Blanks — Angles of Bevel Gear Blanks 102 

Chapter IV. 
Spiral Gears — Calculations for Pitch of Spirals 109 

Chapter V. 

Examples in Calculations of Pitch of Spirals — Angle of 
Spiral — Circumference of Spiral Gears — A few Hints 
on Cutting 113 



CONTENTS. VII 

Chapter VI. 



PAGE. 



Normal Pitch of Spiral Gears — Curvature of Pitch Surface 

— Formation of Cotters 116 

Chapter VII. 
Cutting Spiral Gears in a Universal Milling Machine 122 

Chapter VIII. 
Screw Gears and Spiral Gears — General Remarks 129 

Chapter IX. 

Continued Fractions — Some Applications in Machine Con- 
struction 132 

Chapter X. 
Angle of Pressure 137 

Chapter XI. 
In fcernal Gears — Tables — Index 139 

Chapter XII. 
Strength of Gears — Tables 142 



PART I. 



CHAPTER I. 

PITCH CIRCLE, PITCH, TOOTH, SPACE, ADDENDUM OR FACE, FLANK, 

CLEARANCE. 



Let two cylinders, Fig. 1, touch each other, their . original Cyi 
axes be para" lei and the cylinders be on shalts, turning 
freely. If, now, we turn one cylinder, the adhesion of 
its surface to the surface of the other cylinder will 
make that turn also. The surfaces touching each 
other, without slipping one upon the other, will evi- 
dently move through, the same distance in a given 

. . Linear Veloci 

time. This surface speed is called linear velocity. ty. 
TANGENT CYLINDERS. 




Linear Velocity is the distance a point moves along 
a line in a unit of time. 

The line described by a point in the circumference 
of either of these- cylinders, as it, rotates, may be called 
an arc. The length of the arc (which may be greater 
or less than the circumference of cylinder), described 
in a unit of time, is the velocity. The length, expressed 
in lineal* units, as inches, feet, etc., is the linear velocity. 



EEOWS" t SHAP.PE MEG. CO. 



The length, expressed in angular units, as degrees, is 
the angular Telocity. 

If now, instead of l 3 we take 360 = , or one turn, as 
Angular "v e- t ] le ann^ar nnit, and 1 minute as the time unit, the 
angular velocity will be expressed in turns or revolu- 
tions per minute. 

If these two cylinders are of the same size, one will 

make the same number of turns in a minute that the 

Relative An- other makes. It cne cylinder is twice as large as the 

gular "\ elocitr. _ " ° 

other, the smaller will make two turns while the larger 
makes one. but the linear velocity of the surface of 
each cylinder remains the same. 

This combination would be very useful in mechan- 
ism if we could be sure that one cylinder would always 
turn the other without slipping. 




_-s- 



LancL 

Addendum. 
Tooth. 
Gear. 
Train. 




-:::•.:_>.' 



- ~z- 

LINE 



_~ : : t - 



=?:_£ 



IFi~. 3 



In the periphery of these two cylinders, as in Fig. 
2, cut equidistant grooves. In any grooved piece the 
places between grooves are called lands. Upon the 
lands add part-; ; these parts are called addenda. A 
land and its al.lendum is called a tooth. A toothed 
cylinder is called a gear. Two or more gears with 
teeth interlocking are called a train. A Hue. c c\ Fig. 



PROVIDENCE, R. I 



Addendum 
Circle. 



2 or 3, between the centers of two wheels is called the Line of Cen 
line of centers. A circle just touching the addenda ters ' 
is called the addendum circle. 

The circumference of the cylinders without teeth is 
called the pitch circle. This circle exists geometri- Pltch Circle - 
cally in every gear and is still called the pitch circle pitcll circle 
or the primitive circle. In the study of gear wheels, it th e alS PrinStive 
is the problem to so shape the teeth that the pitch circle, 
circles will just touch each other without slipping. 

On two fixed centers there can turn only two circles, 
one circle on each center, in a given relative angular 
yelocity and touch each other without slipping. 




THICKNESS OF 
WIDTH OF SPACE j TOOTH 

AT PITCH LINE | AT PITCH LINE 




Fig. 4 



4 BROWN & SHARPE MEG. CO. 

Space. The groove between two teeth is called a space. 

In cut gears the width of space at pitch line and thick- 
ness of tooth at pitch line are equal. The distance 
between the center of one tooth and the center of the 

circular Pitch, next tooth, measured along the pitch line, is the cir- 
cular pitch ; that is, the circular pitch is equal to a 
Tooth Thick- tooth and a space; hence, the thickness of a tooth at 

ness. x . 

the pitch line is equal to one-half the circular pitch. 
•^P^^V Let D = diameter of addendum circle. 

tions of Parts 

for Teeth and t< D — " " pitch " 

" P' = circular pitch. 

" t= thickness of tooth at pitch line. 

" s = addendum or face, also length of working 

part of tooth below pitch line or flank. 
" 2s=D" or twice the addendum, equals the 

working depth of teeth of two gears in 

mesh. 
" f— clearance or extra depth of space below 

working depth. 
" s +f= depth of space below pitch line. 
" "D" +f= whole depth of space. 
" N= number of teeth in one gear. 
" 7r=3.1416 or the circumference wlien diameter 

is 1. 
P' is read "P prime/' D" is read "D second." 
n is read, "pi." 

If we multiply the diameter of any circle by tt, the 
To find the product will bs the circumference of this circle. If 
and Diameter we divide the circumference of any circle by tt, the 
quotient will be the diameter of this circle. 



CHAPTER II. 

CLASSIFICATION -SIZING BUNKS AND TOOTH PARTS FROM 
CIRCULAR PITCE— CENTRE DISTANCE— PATTERN GEARS. 



If we conceive the pitch, of a pair of cears to be Elements of 

x * ° the Teeth. 

made the smallest possible, we ultimately come to the 
conception of teetn that are merely lines upon the 
original pitch surfaces. These lines are called ele- 
ments of the teeth. Gears may be classified with 
reference to the elements of their teeth, and also with 
reference to the relative position of their axes or -shafts. 
In most gears the elements of teeth are either straight 
lines or helices (screw-like lines). 

Part I. of this book, treats upon three kinds of 
gears. 

First — Spur Gears ; those connecting parallel shafts ^ tVX GearB - 
and whose tooth elements are straight. 

Second — Bevel Gears; those connecting shafts Bevel Gears, 
whose axes meet when sufficiently prolonged, and the 
elements of whose teeth are straight lines. In bevel 
gears the surfaces that touch each other, without 
slipping, are upon cones or parts of cones whose 
apexes are at the same point where axes of shafts meet. 

Third — Screw or Worm Gears; those connecting w |££ e Jlars. r 
shnfts that are not parallel and do not meet, and the 
elements of whose teeth are helical or screw-like. 

The circular }>itcfi and number of teeth in a wheel giz . n? 
beingf o-iven, the diameter of the wheel and size of Blanks, &<$. 
tooth parts are found as follows : 

Dividing by 3.1410 is the same as multiplying by 
Now - — J— T — 3183; hence, multiply the cir- 



3-14 16- 3-14 16 

cumference of a circle by .3183 and the product will be 
the diameter of the circle. Mult' ply the circular pitch 
by .3183 and the product will be the same part of the 



G BBOWM & SHARPE MFG. CO. 

diameter of pitch circle that the circular pitch is of the 

circumfeivnce of pitch circle. This part or module is 

a Diameter C; ,ii e d a diameter pitch. There are as manv diameter 

Pitch, or Mod- \ 

uie. pitches contained in the diameter of a pitch circle as 

there are teeth in the wheel. 
ai^the^ddeH 6 Most mechanics make the addendum of teeth equal 
?ili^^ a ^f to the module. Hence we can designate the module or 

the >anie. raai- o 

an y- diameter pitch by the same letter as we do the adden- 

dum ; that is, let s=the module. 

.3183 V'=$, or circular pitch multiplied by .3183=5. 
or the module. 

X>=D', or number of teeth in a wheel, multiplied by 

Diameter of ' * J 

Pitch circle, the module, equals diameter of pitch circle. 

(N-j-2) s=D, or add 2 to the number of teeth, mul- 

wiioieDiam-tinv the sum by s and product will be the whole di- 
eter. L - - r 
a meter. 

-=/, or one tenth of thickness of tooth at pitch-line 
clearance. equals amount added to bottom of space for clearance. 
Some mechanics prefer to make/ 1 equal to T *g- of the 
working depth of teeth, or .0625 D'\ One-tenth of 
the thickness of tooth at pitch-line is more than one- 
sixteenth of working depth, being .07S54 D". 
Example. Example. — Wheel 30 teeth. U" circular pitch. 

c . _, . P'=1.5" : then t—~o" or thickness of tooth equals §■". 

Sizes ol Blank J- * 

and Tooth s = 1.5 " x .31 83=.47 • ^> " = moduie for 1^ " P'. (See 

- for Gear 

of so teeth. \% tables of tooth parts, pages 145 -14 s ). 
phcA; 11 D' = 30X.4775" = 14.325' = diaineter of pitch-circle. 

D=(30+2)X.4775"=15.280''=diameter of adden- 
dum circle. 

/ — tV °f -75'' = .075" = clearance at bottom of space. 
'D" = '2x. 4775" = . 9549" = working depth of teeth. 
D''-/-2x.4775'>.075" = 1.0299" = whole depth of 
space. 

s+/=. 4775" -.075" = . 5525" = depth of space inside 
of pitch-line. 

D"=-2s or the working depth of teeth is equal to 
two modules. 

In making calculations it is well to retain the fourth 
place in the decimals, but when drawings are passed 
into the workshop, three places of decimals are suffi- 
cient. 



PROVIDENCE, K. I. 




Fig. 5, Spur Gearing. 



8 BROWN & SHARPE MFG-. CO. 

Distance be- The distance between' the centers of two wheels is 

Iween centers 

of two Gears, evidently equal to the radius of pitch-circle of one wheel 
added to that of the other. The r.idius of pitch-circle 
is equal to s multiplied by one-half the number of teeth 
in the wheel. 

Hence, if we know the number of teeth in two wheels, 
in mesh, and the circular pitch, to obtain the distance 
between centers we first find s ; then multiply s by one- 
half the sum of number of teeth in both wheels and the 
product will be distance between centers. 

Example — "What is the distance between the centers 
of two wheels 35 and 60 teeth, 1J" circular pitch. We 
first find s to be 11" x .3183 = . 3979". Multiplying by 
47.5 (one-half the sum of 35 and 60 teeth) we obtain 
18.899" as the distance between centers. 
Q^T™ f fJT Pattern Gears should be made large enough to 

cnnnKage in o o 

Gear Castings, allow for shrinkage in casting. In cast iron the shrinkage 
is about ijj inch in one foot. For gears one to two feet 
in diameter it is well enough to add simply -j-j-g- of 
diameter of finished gear to the pattern. In gears 
about six inches diameter or less, the moulder will 
generally rap the pattern in the sand enough to make 
any allowance for shrinkage unnecessary. In pattern 
gears the spaces between teeth should be cut wider 
than finished gear spaces to allow for rapping and to 
avoid having too much cleaning to do in order to have 
gears run freely. In cut patterns of iron it is generally 
Metal Pattern enough to make spaces .015" to .02 ' wider. This 
makes clearance .03" to .04" in the patterns. Some 
moulders might want .06" to .07" clearance. 

Metal patterns should be cut straight ; they work 
better with no draft. It is well to leave about .005" to 
be finished from side of patterns after teeth are cut ; 
this extra stock to be taken away from side where 
cutter comes through so as to take out places where 
stock is broken out. The finishing should be done 
with file or emery wheel, as turning in a lathe is likely 
to break out stock as badly as a cutter might do. 

If cutters are kept sharp and care is taken when 
coming through the allowance for finishing is not nec- 
essary and the blanks may be finished before they are 
cut. 



CHAPTER III. 
SINGLE-CURVE GEARS OF 30 TEETH AND OVER. 



Single-curve teeth are so called because they have T g^ le Curve 
but one curve by theory, this curve forming both face 
and flank of tooth sides. In any gear of thirty teeth 
and more, this curve can be a single arc of a circle 
whose radius is one-fourth the radius of the pitch 
circle. In gears of thirty teeth and more, a fillet is 
added at bottom of tooth, to make it stronger, equal 
in radius to one-seventh the widest part of tooth space. 

A cutter formed to leave this fillet has the advantage 
of wearing longer than it would if brought up to a 
corner. 

In gears less than thirty teeth this fillet is made the 
same as just given, and sides of teeth are formed with 
more than one arc, as will be shown in Chapter VI. 

Having calculated the data of a efear of 30 teeth, f Example of a 

, . , fo ' 4 Gear, N=30, P 

inch circular pitch (as we did in Chapter II. for lh" -%"• 
pitch), we proceed as follows : 

1. Draw pitch circle and point it off into parts equal Geometrical 

, , ,„ ., . , .. , Construction. 

to one-hall the circular pitch. Fig. 6. 

2. From one of these points, as at B, Fig. 6, draw 
radius to pitch circle, and upon this radius describe a 
semicircle; the diameter of this semicircle being equal 
to radius of pitch circle. Draw addendum, working 
depth and whole depth circles. 

3. From the point B, Fig. 6, where semicircle, pitch 
circle and outer end of radius to pitch circle meet, lay 
off a distance upon semicircle equal to one-fourth the 
radius of pitch circle, shown in the figure at BA, and 
is laid off as a chord. 

4. Through this new point at A, upon the semicircle, 
draw a circle concentric to pitch circle. This last is 



10 



BROWS k SHAEPE MEG. CO. 



Fig. 6 



r^^SWM 




GEAR, 


30 TEETH, 


"CIRCULAR PITCH, 


P'= 


= for .75" 


N = 


=30 


P 


=4.1888" 


t 


= .375" 


S 


= .2387* 


D' 


= .4775" 


S+/ 


= .2762' 


D'+f 


= .5150" 


D' 


= 7.1610" 


D 


= 7.6384' 



SINGLE CURVE GEAR. 



PROVIDENCE, R. I. 11 

called tlie base circle, and is the one for centers of 
tooth arcs. In the system of single cnrve gears, we 
have adopted the diameter of this circle is .968 of the 
diameter of pitch circle. Thus the base circle of any 
gear 1 inch pitch diameter by this system is .968". 
If the pitch circle is 2" the base circle will be 1.936." 

5. With dividers set to one-quarter of the radius of 
pitch circle, draw arcs forming sides of teeth, placing 
one leg of the dividers in the base circle and letting 
the other leg describe an arc through a point in the 
pitch circle that was made in laying off the parts equal 
to one-half the circular pitch. Thus an arc is drawn 
about A as center through B. 

6. With dividers set to one-seventh of the widest part 
of tooth space, draw the fillets for strengthening teeth 
at their roots. These fillet arcs should just touch the 
whole depth circle and the sides of teeth already 
described. 

Single curve or involute gears are the only gears invori^Gear- 
that can run at varying distance of axes and transmit lng- 
unvarying angular velocity. This peculiarity makes 
involute gears specially valuable for driving rolls or 
any rotating pieces, the distance of whose axes is 
likely to be changed. 

The assertion that gears crowd harder on bearings Pressure on 

° ° bearings. 

when of involute than when of other forms of teeth, 
has not been proved in actual practice. 

Before taking next chapter, the learner should make Practice, De- 
several drawings of gears 30 teeth and more. Say next chapter. 
make 35 and 70 teeth 1|" P'. Then make 40 and 65 
teeth !" F 

An excellent practice will be to make drawing on 
cardboard or Bristol-board and cut teeth to lines, thus 
making paper gears ; or, what is still better, make them 
of sheet metal. By placing these in mesh the learner 
can test the accuracy of his work. 



12 BKOWN & SHAKPE MFG. CO. 



CHAPTER IV. 

RACK TO MESH WITH SINGLE-CURYE GEARS HAYING 
30 TEETH AND OYER. 



made a g re ^ra' This gear (Fig. 7) is made precisely the same as gear 
tory to drawing i n Chapter III. It makes no difference in which direc- 
tion the construction radius is drawn, so far as obtain- 
ing form of teeth and making gear are concerned. 

Here the radius is drawn perpendicular to pitch line 
of rack and through one of the tooth sides, B. A semi- 
circle is drawn on each side of the radius of the pitch 
circle. 

The points A and A' are each distant from the point 
B, equal to one-fourth the radius of pitch circle and 
correspond to the point A in Fig. 6. 

In Fig. 7 add two lines, one passing through B and 
A and one through B and A'. These two lines form 
angles of 75- 10 (degrees) with radius BO. Lines BA 
and BA' are called lines of pressure. The sides of 
rack teeth are made perpendicular to these lines. 
Rack. A Rack is a straight piece, having teeth to mesh 

with a gear. A rack may be considered as a gear of 
infinitely long radius. The circumference of a circle 
approaches a straight line as the radius increases, and 
when the radius is infinitely long any finite part of the 
construction circumference is a straight line. The pitch line of a 

of Pitch Line of . .,-.'. 

Hack. rack, then, is merely a straight line just touching the 

pitch circle of a gear meshing with the rack. The 

thickness of teeth, addendum and depth of teeth 

below pitch line are calculated the same as for a wheel. 

(For pitches in common use, see table of tooth parts.) 

The term circular pitch when applied to racks can be 

more accurately replaced by the term linear pitch. 

Linear applies strictly to a line in general while circular 

pertains to a circle. Linear pitch means the distance 

between the centres of two teeth on the pitch line 

whether the line is straight or curved. 



PROVIDENCE, R. I. 



13 



A rack to mesh with a single-curve gear of 30 teeth 
or more is drawn as follows : 

1. Draw straight pitch line of rack ; also draw ad- 
dendum line, working depth line and whole depth line, 
each parallel to the pitch line (see Fig. 7). 



Rack. 
Fig. 7. 




RACK TO MESH WITH SINGLE CURVE GEAR 
HAVING 30 TEETH AND OVER. 



14 BROWN i: SHARPE MFG. CO. 

2. Point off the pitch line into parts equal to one- 
half the circular pitch, or = t. 

3. Through these points draw lines at an angle of 
7-H with pitch lines, alternate lines slanting in oppo- 
site directions. The left-hand side of each rack tooth 
is perpendicular to the line BA. The right-hand side 
of each rack tooth is perpendicular to the line BA'. 

4. Add fillets at bottom of teeth equal to 4- of the 
width of spaces between the rack teeth at the adden- 
dum line. 

sid\? g of e nick Tlie sketch, Fig. 8, will show how to obtain angle cf 
Teeth. sides of rack teeth, directly from pitch line of rack, 

without drawing a gear in mesh with the rack. 




Upon the pitch line b b', draw any semicircle — 
b a a' 1/ . From point b lay off upon the semicircle 
the distance b «, equal to one-quarter of the diameter 
of semicircle, and draw a straight line through b and a. 

This line, b a, makes an angle of 75.}° with pitch line 
bb', and can be one side of rack tooth. The same 
construction, b' a', will give the inclination 75|° in the 
opposite direction for the other side of tooth. 

The sketch, Fig. 9, gives the angle of sides of a tool 
for planing out spaces between rack teeth. Upon any 
line OB draw circle OABA'. From B lay off distance 
BA and BA'. each equal to one-quarter of diameter of 
the circle. 

Draw lines OA and OA . These two lines form an 
angle of 29°, and are right for inclination of sides of 
rack tool. 



PROVIDENCE, R. I. 



15 



Make end of rack tool .31 of circular pitch, and then ^Y 11 * 1 of ? ack 

■t ' Tool at end. 

round the corners of the tool to leave fillets at the 
bottom of rack teeth. 

Thus, if the circular pitch of a rack is 1J" and we 
multiply by .31, the product .465" will be the width of 
tool at end for rack of this pitch before corners are 
taken off. This width is shown at x y. 




A Worm is a screw that meshes with the teeth of a 
gear. 

This sketch and the foregoing rule are also right for Worm Thread 
a worm-thread tool, but a worm-thread tool is not 
usually rounded for fillet. In cutting worms, leave 
width of top of thread .335 of the circular pitch. 
When this is done, the depth of thread will be right. 




SKETCH OF WORM THREAD 



16 



CHAPTER V. 

DIAMETRAL PITCH— SIZING BLANKS AND THE TEETH OF SPUR GEARS 
—DISTANCE BETWEEN THE CENTRES OF WHEELS. 



In making drawings of gears, and in cutting racks, 

When it is . . ° ° 7 - , 

necessary to it is necessary to know the circular pitch, both on 

know the Cir- . " . . . 

cuiar Pitch. account oi spacing teeth and calculating then* strength. 

It would be more convenient to express the circular 

pitch in whole inches, and the most natural divisions 

in a complete of an inch, as 1" P', f" P', \'' P', and so on. But as 

Pitch circum- the circumference of the pitch circle must contain the 

ference must • i ■. i i i -, m • ■ 

contain the cir- circular pitch soine whole number ot times, corre- 
some whole sponding to the number of teeth in the gear, the 
times ber ° diameter of the pitch circle will often be of a size not 
readily measured with a common rule. This is because 
the circumference of a circle is equal to 3 1416 times 
the diameter, or the diameter is equal to the circum- 
ference multiplied by .3183. 

In practice, it is better that the diameter should be 
Pitch, j .n f some size conveniently measured. The same applies 

Terms ot the ■ L L 

Diameter. fc the distance between centers. Hence it is generally 
more convenient to assume the pitch in terms of the 
diameter. In Chapter II. was given a definition of a 
diameter pitch, and also how to get a diameter pitch 
from the circular pitch. 

"We can also assume a diameter pitch and pass to its 
circular Pitch equivalent circular pitch. 11 the circumference of the 

and a JJiame- x x 

ter Pitch. pitch circle is divided by the number of teeth in the 

gear, the quotient will be the circular pitch. In the 
same manner, if the diameter of the pitch circle is 
divided by the number of teeth, the quotient will be a 
diameter pitch. Thus, if a gear is 12 inches pitch 
diameter and has 48 teeth, dividing 12" by 43, the 
quotient J" is a diameter pitch of this gear. In prac- 



PROVIDENCE, R. I. 17 

tice, a diameter pitch is taken in some convenient part 

of an inch, as A-'' diameter pitch, and so on. It ^Abbreviation 

' ^ t * ' of Diameter 

is convenient in calculation to designate one of these Pitch, 
diameter pitches by s, as in Chapter II. Thus, for \" 
diameter pitch, s is equal to J". Generally, in speak- 
ing of diameter pitch, the denominator of the fraction 
only is named. J" diameter pitch is then called 3 
diametral pitch. That is, it has been found more con- 
venient to take the reciprocal of a diameter pitch in 
making calculation. The reciprocal of a number is 1, 
divided by that number. Thus the reciprocal of J is a NuE? 
4, because \ goes into 1 four times. 

Hence, we come to the common definition : 
Diametral Pitch is the number of teeth to one inch pitch. 
of diameter of pitch circle. Let this be denoted by P. 
Thus, J" diameter pitch we would call 4 diametral 
pitch or 4 P, because there would be 4 teeth to every 
inch in the diameter of pitch circle. The circular 
pitch and the different parts of the teeth are derived 
from the diametral pitch as follows. 

^iifi = P', or 3.1416 divided by the diametral pitch Given the Di- 

r i/ x ametral to find. 

is equal to the circular pitch. Thus to obtain the cir-*£ e Circular 
cular for 4 diametral pitch, we divide 3.1416 by 4 and 
get .7854 for the circular pitch, corresponding to 4 c £^ ta pitoh 
diametral pitch. traiPit?n. ame " 

In this case we would write P=4,P' = .7854", *=i". 
£-"— s, or one inch divided by the number of teeth to 
an inch, gives distance on diameter of pitch circle 
occupied by one tooth. The addendum or face of 
tooth is the same distance as s. 

i _ 



s 



=P, or one inch divided by the distance occupied 



by one tooth equals number of teeth to one inch. 

±*l—t t or 1.57 divided by the diametral pitch gives amJSprtchto 
thickness of tooth at pitch line. Thus, thickness of JJes^of Tooth 
teeth along the pitch line for 4 diametral pitch is .392". L* n e he Pitch 

^=D', or number of teeth in a gear divided by the N umber *o? 
diametral pitch equals diameter of the pitch circle, and the a Diam- 
Thus for a wheel, 60 teeth, 12 P, the diameter of^h^Diam- 
pitch circle will be 5 inches. gg^* Pitch 

— i^=D, or add 2 to the number of teeth in a wheelN umber of 

_. -1-1 -i Teeth ina wheel 

and divide the sum bv the diametral pitch, and the and the Diame- 

J r tral Pitch to 

find the Wholo 
Diameter. 



18 BKOWN & SHARPE MFG. CO. 

quotient will be the whole diameter of the gear or the 
diameter of the addendum circle. Thus, for 60 teeth, 
12 P, the diameter of gear blank will be 5fv inches. 

£,=P, or number of teeth divided by diameter of 
pitch circle in inches, gives the diametral pitch or 
number of teeth to one inch. Thus, in a wheel, 24 
teeth, 3 inches pitch diameter, the diametral pitch is 8. 

— jp = P, or add 2 to the number of teeth; divide the 
sum by the whole diameter of gear, and the quotient 
will be the diametral pitch. Thus, for a wheel 3^'' 
diameter, 14 teeth, the diametral pitch is 5. 

D' P=N, or diameter of pitch circle, multiplied by 
diametral pitch equals number of teeth in the gear. 
Thus, in a gear, 5 pitch, 8" pitch diameter, the num- 
ber of teeth is 40. 

D P — 2=N or multiply the whole diameter of the 
gear by the diametral pitch,subtract 2, and the remain- 
der will be the number of teeth. 

j^p = s, or divide the whole diameter of a spur gear 
by the number of teeth plus two, and the quotient 
will be the addendum, or a diameter pitch. 

" Pitch >iameter ^ n f u ^ ure ? "vvhen we speak of a diameter pitch, we 

shall mean the addendum distance or s. If we speak 

of so many diameter pitches, we shall mean so many 

The Diame- times s, (-3= s). "When we say the diametral pitch we 

tral Pitch. F ' - J * 

shall mean the number of teeth to one inch of diameter 
of pitch circle, or P, (— =P). 



s 



To obtain Di- 



Pitc When the circular pitch is given, to find the corre- 
pitch Circular sponding diametral pitch, divide 3.1416 by the circular 
pitch. Thus 1.57 P is the diametral pitch correspond- 
ing to 2-inch circular pitch, (^l^i^=P). 
Example. What diametral pitch corresponds to J-" circular 

pitch ? Remembering that to divide by a fraction we 
multiply by the denominator and divide by the numer- 
ator, we obtain 6.28 as the quotient of 3.1416 divided by 
J . 6.28 P, then, is the diametral pitch corresponding 
to J circular pitch. This means that in a gear of \ 
inch circular pitch there are six and twenty-eight one 
hundredths teeth to every inch in the diameter of the 
pitch circle. In the table of tooth parts the diametral 



PROVIDENCE, R. I. 19 

pitches corresponding to circular pitches are carried 
out to four places of decimals, but in practice three 
places of decimals are enough. 

When two gears are in mesh, so that their pitch 
circles just touch, the distance between their axes or 
centers is equal to the sum of the radii of the two 
gears. The number of the diameter pitches between 
centers is equal to half the sum of number of teeth in 
both gears. This principle is the same as given in 
Chapter II., page 6, but when the diametral pitch an d -o^nce ^ht 
numbers of teeth in two gears are given, add together tween centers. 
the numbers of teeth in the two wheels and divide half 
the sum by the diametral pitch. The quotient is the 
center distance. 

A gear of 20 teeth, 4 P, meshes with a gear of 50 Example, 
teeth : what is the distance between their axes or 
centers ? Adding 50 to 20 and dividing half the sum 
by 4, we obtain 8f" as the center distance. ' 

The term diametral pitch is also applied to a rack. 
Thus, a rack 3 P, means a rack that will mesh with a 
gear of 3 diametral pitch. 

It will be seen that if the expression for a diameter Fractional 

. -,, , \ Diametral 

pitch has any number except 1 for a numerator, we Pitch, 
cannot express the diametral pitch by naming the 
denominator only. Thus, if the addendum or a diam- 
eter pitch is f^, the diametral pitch will be 2^, because 
1 divided by T \ equals 2£. 

In Chapter II, the term modtde is used in the same 
sense as the term a diameter pitch. Modtde is much 
used where gears are made to metric sizes, for the 
reason that, the millimeter being so short, the module 
is conveniently expressed in millimeters. If we know 
the module of a gear we can figure the other parts as 
easily as we can if we know either the circular pitch 
or the diametral pitch. The module is, in a sense, an 
actual distance, while the diametral pitch, or the num- 
ber of teeth to an inch, is a relation or merely a ratio. 
The meaning of the module is not easily mistaken. 



20 EEOWU i SHiEPE 5IT(i. CO. 



CHAPTER VI. 

SUGLE-GURYE GEARS HAYING LESS THAN 30 TEETH— GEARS AND 
RACKS TO MESH WITH GEARS HAVING LESS THAN 30 TEETH. 



Construction, Jq Yig. 10. tlie construction of the rack is the same 

Fig. 10. ° 

as the construction of the rack in Chapter TV. Tne 
gear in Fig. 10 is drawn from base circle out to adden- 
dum circle, by the same method as the gear in Chapter 
III., but the spaces inside of base circle are drawn as 
follows : 
Flanks of T n gears, 12 to 19 t-eth, the sides of spaces inside 

Gears in low = ' ' " 

numbers of,,f the base circle are radial for a distance, a b, equal 

Teeth. «, - » ' l 

to ^, or 3.5 divided by the product of the pitch by the 
number of teeth. In gears with more than 10 teeth 
the radial construction is omitted. 
construction Then, with one leg- of dividers in pitch circle in 

of Fig. 10 con- ° * 

tinued. center of next tooth, e, and other leg just touching 

one of the radial lines at l>. continue the tooth side 
into c, until it will touch a fillet arc. whose radius is 
i the width of space at the addendum circle. The 
part, V c', is an arc from center of tooth f/, etc. The 
flanks of teeth or spaces in gear, Fig. 11, are made the 
same as those in Fig. 10. 

This rule is merely conventional or not founded 
upon any principle other than the judgment of the de- 
signer, to efTe:-t the object to have spaces as wide as 
practicable, just below or inside of base circle, and 
then strengthen flank with as large a fillet as will clear 



& 



addenda of any gear. If flanks in any gear will clear 
addenda of a rack, they will clear addenda of any 
Internal Gear, other gear, except internal gears. An internal gear is 
one having teeth upon the inner side of a rim or ring. 
Now, it will be seen that the gear, Fig. 10, has teeth 



PROVIDENCE, R. I. 



21 




Fig. 10 



22 



re. :w>- i snAzrz irre. ::. 



too much rounded at the points or at the addendum 

arcle. In gears cf pitch coarser than 10 to r_ch (10 

AddTSof ^)» an( ^ having less than 30 teeth, this rounding 

1 - — becomes obj ectionable. This rounding occur s, be a nse 

in these gears arcs of en : les depart too far from the 

true involute carve, being so much that points of 

teeth get no bearing on flanks of teeth in other wheels. 

In gear. Fir 11. r"_e teetii outside of base circle are 

made as nearly true involute as a workman will be able 

t:- ret vri:L: : =^ e: ;.l n; :_r_r.~. This is accomplished 

Ajproxtma-ag follows: draw three or four tan rents to the base 

Don : _rue jz.- o 

volute, circle, * t", jj\ k k\ 1 1', letting tie points of tangen sy 

on base circle i'.j\ k', V be about ^ or J the circular pitch 
apart : the first point, *', being distant from i, equal to 
| the radius of pitch circle. "With dividers set to -J- 
the radius of pitch circle, placing one leg in /', draw 
the arc, a' * j/ with one leg in j' 7 and radius j' j, 
dxawj U; with one leg in k . and radius I: k draw J: 7. 
Should the addendum circle be outside of 7. the I : oth 
side can be completed with the last radius, I' /. The 
arcs, a ij, j k and k l. together form a very close 
approximation to a true involute from the base circle, 
i' j' k' I'. The exact involute for gear teeth is the 
curve made by the end of a band when unwound from 
a cylinder of the same diameter as base circle. 

The foregoing operation of drawing tooth sides, 
although te 1: : - in les iription. is very easy of practical 
application. 
Bounding of It will also be seen that the addenda of rack teeth 

A i '- r z. _ i : : _ 

Rack. in Tig. 10. interfere with the gear-teeth nanks, as ;.: 

m n; to avoid this interference, the teeth of rack, Fig. 
11. are rounded at points cr addenda- 
It is also necessary to round off the points of invo- 
lute teeth in high numbered gears, when they are to 
interchange with low nunir. e: ed _ e ~ In iniercshang - 
able sets of gears the lowest-numbered pinion is usual- 
Temple:? iv 12. Just how much to round off can be 1 earned by 
K::i::i. : _ - -- ■-.'-•- - ----_■- -~ -; a Zzii:'^ :n: :: :_:n met.-- cr 

Points of teeth. ° L 

..dboard. for the gear and rack, or, two gears re- 
quired, and fitting addenda of teeth to clear flar k - 
However accurate we may make a diagram, it is qu 



mOVIDENCE, E. I. 



23 




Fig. 



24 



BROWN ft SHARPE MFG. CO. 



as well to make templets in order to shape cutters 
accurately. 

It is best to make cutters to corrected diagrams, as 
in Fig. 11. TVhen corrected diagrams are made, as 
in Fig. 11. take the following: 

For 12 and 13 teeth, diagram of 12 teeth. 



a Set of Cut- 
ters. 



14 


to 16 


17 


" 20 


21 


" 25 


26 


" 34 


35 


" 54 


55 


" 134 


135 


" rack, 



tc 


; i 


14 


a 


ti 


a 


17 


ic 


tc 


i. i 


21 


it 


(< 




26 


i i 


It 


a 


35 


a 


hi 


t i 


55 




i . 


-135 


i i 



Templets for large gears must be fitted to run with 
12 teeth. 



PROVIDENCE, R. I. 



25 



CHAPTER VII. 
DOUBLE-CURYE TEETH— GEAR, 15 TEETH— RACK. 



In double-curve teeth the formation of tooth sides ah Double- 
curve Tooth 

changes at the pitch line. In all gears the part of Fa °es are con- 
teeth outside of pitch line is convex ; in some gears 
the sides of teeth inside pitch line are convex ; in some, 
radial; in others, concave. Convex faces and concave 
flanks are most familiar to mechanics. In interchange- 
able sets of gears, one gear in each set, or of each 
pitch, has radial flanks. In the bast practice, this gear 
has fifteen teeth. Gears with more than fifteen teeth, 
have concave flanks ; gears with less than fifteen teeth, 
have convex flanks. Fifteen teeth is called the Base 
of this system. 

We wid first draw a gear of fifteen teeth. This construction 

° of Fig. 12. 

fifteen-tooth construction enters into gears of any 
number of teeth and also into racks. Let the gear be 
3 P. Having obtained data, we proceed as follows : 

1. Draw pitch circle and point it off into parts equal 
to one-thirtieth of the circumference, or equal to thick- 
ness of tooth =i^. 

2. From the center, through one of these points, as 
at T, Fig. 12, draw line OTA. Draw addendum and 
whole-depth circles. 

3. About this point, T, with same radius as 15-tooth 
pitch circle, describe arcs A K and k. For any other 
double-curve gear of 3 P., the radius of arcs, A K and 
O k, will be the same as in this 15-tooth gear=2J". 
In a 15-tooth gear, the arc, O k, passes through the 
center O, but for a gear having any other number of 
teeth, this construction arc does not pass through 
center of gear. Of course, the 15-tooth radius of arcs, 
A K and O k, is always taken from the pitch we are 
working with. 



26 



BROWN k SHAEPE MTG. CO. 




EiS- 12 

DOUBLE CURVE GEAR. 



PROVIDENCE, R. I. 27 

4. Upon these arcs on opposite sides of line OTA, 
lay off tooth thickness, A K and O k, and draw line 
KT&. 

5. Perpendicular to K T k, draw line of pressure, 
L T P ; also through O and A, draw lines A R and O r, 
perpendicular to K T &. The line of pressure is at 
an angle of 78° with the radius of gear. 

6. From 0, draw a line R to intersection of AR 
with KT^. Through point c, where R intersects 
L P, describe a circle about the center, 0. In this 
circle one leg of dividers is placed to describe tooth 
faces 

7. The radius, c d, of arc of tooth faces is th*. 
straight distance from c to tooth-thickness point, b, 
on the other side of radius, O T. With this radius, c b, 
describe both sides of tooth faces. 

8. Draw flanks of all teeth radial, as e and Of 
The base gear, 15 teeth only, has radial flanks. 

9. With radius equal to one-seventh of the widest 
part of space, as g h, draw fillets at bottom of teeth. 

The foresroiner is a close approximation to epicy- . Approxima- 

° *=> rr r J tion to Epicy- 

cloidal teeth. To get exact teeth, make two 15-tootheioidai Teeth. 
gears of thin metal. Make addenda long enough to 
come to a point, as at n and q. Make radial flanks, as 
at m and p, deep enough to clear addenda when gears 
are in mesh. First finish the flanks, then fit the long 
addenda to the flanks when gears are in mesh. 

When these two templet gears are alike, the centers standard 
are the right distance apart and the teeth interlock 
without backlash, they are exact. One of these tem- 
plet gears can now be used to test any other templet 
gear of the same pitch. 

Gears and racks will be right when they run cor- 
rectly with one of these 15-tooth templet gears. Five 
or six teeth are enough to make in a gear templet. 

Double- curve Rack. — Let us draw a rack 3 P. R^^ig^ST 9 
Having obtained data of teeth we proceed as follows : 

1. Draw pitch line and point it off in parts equal 
to one- half the circular pitch. Draw addendum and 
whole-depth lines. 

2. Through one of the points, as at T, Fig. 13, draw 
line OTA perpendicular to pitch line of rack. 



28 



BBOWS s 8HABPE MFG. CO. 




ZFi s . 13 

DOUBLE CURVE RACK. 



PROVIDENCE, R. I. 



29 



3. About T make precisely the same construction as 
was made about T in Fig. 12. That is, with radius of 
15-tooth pitch circle and center T draw arcs Jc and 
A K ; make O k and A K equal to tooth thickness ; 
draw KT/;; draw r, A R, and line of pressure, each 
perpendicular to K T k. 

4. Through R and r, draw lines parallel to A. 
Through intersections c and c' of these lines, with 
pressure line L P, draw lines parallel to pitch line. 

5. In these last lines place leg of dividers, and draw 
faces and flanks of teeth as in sketch. 

6. The radius c' cV of rack-tooth faces is the same 
length as radius c d of rack-tooth flanks, and is the 
straight distance from c to tooth-thickness point b on 
opposite side of line A. 

7. The radius for fillet at bottom of rack teeth is 
equal to i- of the widest part of tooth space. This 
radius can be varied to suit the judgment of the 
designer, so long as a fillet does not interfere with 
teeth of engaging gear. 




Fig. 14 



Racks of the same pitch, to mesh with interchange- 
able gears, should be alike when placed side by side, 
and fit each other when placed together as in Fig. 14. 

In Fig. 13, a few teeth of a 15-tooth wheel are shown 
in mesh with the rack. 



30 



CHAPTER VIII. 

DOUBLE-CURVE SPUR GEARS, HAYING MORE AND FEWER THAN 
15 TEETH— ANNULAR GEARS, 



Consxracvion ^et us draw two gears, 12 and 24 teeth, 4 P, in 

Ol rig. Id. d 

mesh. In Fig. 15 the construction lines of the lower 
or 24-tooth gear are full. The upper or 12-tooth gear 
construction lines are dotted. The line of pressure, 
L P. and the line K T Jc answer for both gears. The 
arcs A K and O k are described about T. The radius 
of these ares is the radius of pitch circle of a gear 15 
teeth 4 pitch. The length of arcs A K and O k is the 
tooth thickness for 4 P. The line K T k is obtained 
the same as in Chapter YH. for all double-curve gears, 
the distances only varying as the pitch. Having drawn 
the pitch circles, the line K T Jc. and, perpendicular to 
K T Jc, the lines A Pi. r and the line of pressure 
L T P. we proceed with the 24-tooth gear as follows : 

1. From center C. through r, draw line intersecting 
line of pressure in m. Also draw line from center C 
to E. crossing the line of pressure L P at <:■. 

2. Through m describe circle concentric with pitch 
circle about C. This is the circle in which to place 
one leg of dividers to describe flanks of teeth. 

3. The radius, m n. of flanks is the straight distance 
from m to the first tooth-thickness point on other side 
of hue of centers. C C". at v. The arc is continued to 
n. to show how constructed. This method of obtain- 
ing radius of double-curve tooth flanks applies to all 
gear's having more than fifteen teeth. 

4. The construction of tooth faces is similar to 15- 
tooth wheel in Chapter VII. That is : Draw a circle 
through c concentric to pitch circle : in this circle 
place one leg of dividers to draw tooth faces, the 
radius of tooth faces being c b. 



PROVIDENCE, R. I. 



31 



\\ 


\ 


\ 


\ 




\ \ 


/v, 


\ \ 


•^ A 




\\ 


A \ 


> — \\ 

A y\ 


/ \ \ 

\ 
\ 




PINION, 12 TEETH, 
GEAR 24 TEETH, 4 P. 

P=4 

N=12 and 24. 
P'= .7854" 
t = .3927" 
S = .2500" 
D 1 ^ .5000" 
s+ / = .2893" 
D"+/ = .5393" 



D'= 3" 



Id= 

( D'= 6' 



C< 



DOUBLE CURVE GEARS IN MESH. 



32 zzz^zs - t =ez_zzz zzzz ::. 



~ "~ T . 



". rhc : lira :z z_t:s it roots of ---~z is equal to 
no-seventh the width of space at addendum circle. 

The constructions for flanks of !■"-. !•: :-—'. 14 



line :: pressure in u. Tizrzngh u draw circle about 

C. Iz :zis circle :ze leg :: .".-—_ lers :> zlzcei zdt 
iz \~ zzz z.zil_:5. 

'2 __e rzzizz :: zzzizis is :ze 5zz:z:_:e zzzzz z ::> 
:Lt ziz: : :zzz.- :z iczzzc s s r :izz --. ;:.:ir,- z :.'z o: 
CTC. Tzis zi _ r^ ::i _ r: zlzz/zs. Tiie zc :~ c:z- 
:i_.e . :: •. :: ~z:~ czzszrzctizz. 

3. This arc f:r zzzzzis i= : miizzzei z :: z — zr '.. ~.z~ 
zzzter. :zlj a":: : z: :ze siszz ;: :ze — zrlczzz zerzz :: 
J s.) ; the lower part of Hank is similar to flanks of 
gear in Chapter VI. 

Ciizzter VII.. zzz ::■ :ze Ji-t::zz gear iz. zze fire- 
going, Hie I lira zemg to y ; the arc is continued to a^ 



ATJiz^r ; rirs. _Ajz-zz_z. Gziez Gears with teeth inside of a rim 
:r zizg zze cailez \~ r -zlar :r Iz.:einzl Grzrs. TJze 
construction of tooth outlines is similar to the f ore- 

g: iz.gr. c':: :ze z//::z5 :: z sjzzr ez:e:zizl zez: ::e::zze 
tze :-":"" :: :.z e_.r"rizr gezz. 

Prof. MacCord has shown that in the system just 
ze=::r*:e~. :ze z:zi:z zirsizzrg — _:z ez ar_zzziar .ez' 
zizz: :li±er zrzrz i: cv :' '. .-: :::tt1 Zr:'_. Tlzz 
a gear :: -- teeth zz:: work with an annular gear 
of 36 teeth, but it will work with annular gears of 39 
tee:b and more. The n'.ers at the roots of the teeth 
mzs: re •: z I e s.s re'ivs :z..z iz :riiz:-.:v srzr g:ars. .An 
az:: z : ge.-.r : frrizz .zm :ts rz..:e cr i.-=s iz.z 15 
tee: e z: r. This ^vill be shown in Part IL 

Azzzziaz-gezz y.atreizis rerzzie zz:re jzzziz :':: 

zzz'zizi^gtzazz ezrerzai or z z gears 
pzz:_=. In speaking :: linereni -siz ed gears, the smallest 

Tze zze of pressure iz. all z^:^^ except involute, 
eonstaiitlj :L i_ges. 78" is the pressure angle in 
double-curve, or epicrcloidal gears foe an instant 



PROVIDENCE. R. I. 



33 



only; in our example, it is 78° when one side of a 
fcooth reaches the line of centers, and the pressure 
against teeth is applied in the direction of the arrows. 

The pressure angle of involute gears does not 
change. An explanation of the term angle of pressure 
is given in Part II. 

We obtain the forms for ejncycloidal gear cutters 
by means of a machine called the Odontom Engine. 
This machine will cut original gears with theoretical 
accuracy. 

It has been thought best to make 24 gear cutters 24 Double- 

° ° curve Gear 

for each pitch. This enables us to fill any require- Cu "era for 

± j i. each Pitcb. 

ment of gear-cutting very closely, as the range covered 
by any one cutter is so small that it is exceedingly near 
to the exact shape of all gears so covered. 

Of course, a cutter can be exactly right for only one 
gear. Special cutters can be made, if desired. 





1 PITCH TOOTH CURVES 

from the 
ODONTOM ENGINE. 



34 



CHAPTER IX. 

BEVEL-GEAR BLANKS, 



Bevel Gears connect shafts whose axes meet when 
Be ^ J« Qg ai ? f sufficiently prolonged. The teeth of bevel gears are 



cones. 



formed upon formed about the frustrums of cones whose apexes 

frustrums of l 

are at the same point where the shafts meet. In Fig. 
16 we have the axes A O and B 0, meeting at O, and 
the apexes of the cones also at O. These cones are 
called the pitch cones, because they roll upon each 
other, and because upon them the teeth are pitched. 
If, in any bevel gear, the teeth were sufficiently pro- 
longed toward the apex, they would become infinitely 
small ; that is, the teeth would all end in a point, or 
vanish at 0. We can also consider a bevel gear as 
beginning at the apex and becoming larger and larger 
as we go away from the apex. Hence, as the bevel 
gear teeth are tapering from end to end, we may say 




BEVEL GEAR PITCH CONES. 



Fig. 16. 



that a bevel gear has a number of pitches and pitch 
circles, or diameters : in speaking of the pitch of a 
bevel gear, we mean always the pitch at the largest 



PROVIDENCE, R. I. 



35 



pitch circle, or at the largest pitch diameter, as at 
b d, Fig. 17. 

Fig. 17 is a section of three bevel gears, the gear 
o B q being twice as large as the two others. The 
outer surface of a tooth as m m' is called the face of Construction 

of Bevel Gear 

the tooth. The distance m m' is usually called the B1 anks. 
length of the face of the tooth, though the real length 
is the distance that it occupies upon the line O i. The 
outer part of a tooth at m n is called its large end, and 
the inner part m' n' the small end. 

Almost all bevel gears connect shafts that are at 
right angles with each other, and unless stated other- 
wise we always understand that they are so wanted. 

The directions given in connection with Fig. 17 
apply to gears with axes at right angles. 

Having decided upon the pitch and the numbers of 
teeth : — 

1. Draw centre lines of shafts, A O B and COD, 
at right angles. 

2. Parallel to A O B, draw lines a b and c d, each 
distant from A B, equal to half the largest pitch 
diameter of one gear. For 24 teeth, 4 pitch, this half 
largest pitch diameter is 3" '. 

3. Parallel to COD, draw lines e f and g h, dis- 
tant from COD, equal to half the largest pitch 
diameter of the other gear. For a gear, 12 teeth, 4 
pitch, this half largest pitch diameter is 1 J ff . 

4 At the intersection of these four lines, draw 
lines O i, O j, O k, and 1 ; these lines give the size 
and shape of pitch cones. We call them " Cone Pitch 
Lines." 

5. Perpendicular to the cone- pitch lines and through 
the intersection of lines a b, c d, e f , and g h, draw 
lines m n, o p, q r. We have drawn also u v to show 
that another gear can be drawn from the same diagram. 
Four gears, two of each size, can be drawn from this 
diagram. 

6. Upon the lines m n, o p, q r, the addenda and 
depth of the teeth are laid off, these lines passing 



36 



BROWN i- SHARPE MFG. CO. 



through the largest pitch, circle ;: the gears, Layoff 
the s id lentlnm. it beins; in these 2:ears 4- . This sives 
distance m n. o p, q r. and a v equal to the working 
depth of teeth, which in these gears is J . The 
addendum of course is measured perpendicu'.aily from 
the cone pitch lines as at k r. 

7. Draw lines m, n, p, o. . r. 
These lines _ the height of teeth above the cone- 

:teh lines as they approach O, and would vanish 
entirel '. It is quite us well never to have the 

length of teeth, or face, m n longer than one-third 
th- apex distance m 0, nor more than two and one- 
half times the circular pitch 

8. Having decided upon the length of face, draw 
limiting lines in n perpendicular :: i 0. q r perpen- 
dicular to k O. and so on. 

TL- listance between the cone-pitch lines at the 

inner ends of the teeth m n and q r is called the inner 

or smaller pitch diameter, and the circle at these points 

is called the smallest pitch circle. We now Lave the 

outline ;: a -rtion of the gears through their axes. 

The listance m i is the whole diameter of the pinion. 

;f el ^ hol f The distance q o is the whole diameter of the gear. 

- In practice these diameters can be obtained bvmeasur- 

obtained by i n2 r th c Irawing. The diameter of pinion is 3.45 and 

Measuring c - x 

Drawings. " f - r g eai __ We can find the angles also by 

measuring the drawing with a protractor. In the 
absence of a protrack r, templetes can be cut to the 
drawing. The angle form . line m m with a b is 

the angle of face of pinion, in this pinion o'j c 11 . or 
59^° nearly. The lines q and g h give us angle of 
face of gear, for this gear 22 z 19 . or '2'2^ = nea:~. 
The angle formed by m n with a b is called the angle 
of edge of pinion, in our sketch 2»3 : 34', or about 26J : 
The angle of edge of gear, line q r with g h, is 
or about 63J°. In turning blanks to these angles we 
place one arm of the protractor or templet against the 
end of the hub, when trying angles of a blank. Some 
designers give the ausrJes from the axes of gears, bat 



PROVIDENCE, E. I. 



37 




Fig. 17 



38 BE0WX & SHARPE MFG. CO., 

it is not convenient to- try blanks in this way. The 
method that we have given comes right also for angles 
as figured in compound rests. 

When axes are at right angles, the sum of angles 
of edge in the two gears equals 90°, and the sums of 
angle of edge and face in each gear are alike. 

The angles of the axes remaining the same, all pairs 
of bevel gears of the same ratio have the same angle 
of edge ; all pairs of same ratio and of same numbers 
of teeth have the same angles of both edges and faces 
independent of the pitch. Thus, in all pairs of bevel 
gears having one gear twice as large as the other, with 
axes at right angles, the angle of edge of large gear 
is 63° 26 , and the angle of edge of small ffear is 26° 34'. 

> DO G 

In all pairs of bevel gears with axes at right angles, 
one gear having 24 teeth and the other gear having 12 
teeth, the angle of face of small gear is 59° 11'. 
4, n 9 t ] ie , r The following method of obtaining the whole diam- 

rnethod ot ob- c s 

taining Whole ^ er f bevel gears is sometimes preferred : 

Diameter ot c l 

Blanks. From k lay off : upon the cone-pitch line, a distance 

K w, equal to ten times the working depth of the 

teetk=10D". Xow add ts of the shortest distance 

of w from the line g h, which is the perpendicular 

dotted line w x. to the outside pitch diameter of gear, 

and the sum will be the whole diameter of gear. In 

the same manner fu of w y. added to the outside pitch 

diameter of pinion, gives the whole diameter of pinion. 

The part added to the pitch diameter is called the 

» 

diameter increment. 

Part II gives trigonometrical methods of figuring 

bevel gears : in our Formulas in Gearing there arc 

trigonometrical formulas for bevel gears, and also 

tables for angles and sizes. 

Construction ^ somewhat similar construction will do for bevel 

of Bevel-Gear 

Blanks whose gears whose axes are not at right angles. 

Axes are not ° ct ^ 

at Right An- J n Fig. 18 the axes are shown at O B and D, the 
angle BOD being less than a right angle. 

1. Parallel to B. and at a distance from it equal 
to the radius of the gear, we draw the lines a b and c d. 



PROVIDENCE, R. I. 



39 




INSIDE BEVEL GEAR 

AND PINION. Fig.no 



40 



BROWN & SHARPE MFG. CO. 

2. Parallel to O D, and at a distance from it equal 
to the radius of the pinion, we draw the lines e f and g h- 

3. Now, through the point j at the intersection of 
c d and g h, we draw a line perpendicular to O B. 
This line k j, limited by a b and c d, represents the 
largest pitch diameter of the gear. 

Through j we draw a line perpendicular to D. 
This line j 1, limited by e f and g h, represents the 
largest pitch diameter of the pinion. 

4. Through the point k at the intersection of a b 
with k j, we draw a line to O, a line from j to 0, and 
another from 1, at the intersection j 1 and e f to 0. 
These lines O k, O j, and O 1, represent the cone- 
pitch lines, as in Fig. 17. 

5. Perpendicular to the cone-pitch lines we draw 
the lines u v, op, and q r. Upon these lines we h"ry 
off the addenda and working depth as in the previous 
figure, and then draw lines to the point O as before. 

By a similar construction Figs. 19 and 20 can be 
drawn. 




STOCKING CUTTER. 



PROVIDENCE, K. I. . 41 



CHAPTER X. 

BEVEL GEARS. 

FORMS AND SIZES OF TEETH. 

CUTTING TEETH. 

To obtain the form of the teeth in a bevel gear we Form of 

bevel g e a r 

do not lay them out upon a pitch circle, as we do in a teeth, 
spur gear, because the rolling pitch surface of a bevel 
gear, at any point, is of a longer radius of curvature 
than the actual radius of a pitch circle that passes 
through that point. Thus in Fig. 21, let f g c be a 
cone about the axis O A, the diameter of the cone 
being f c, and its radius g c. Now the radius of 
curvature of the surface, at c, is evidently longer than 
g c, as can be seen in the other view at C ; the full 
line shows the curvature of the surface, and the dotted 
line shows the curvature of a circle of the radius g c. 
It is extremely difficult to represent the exact form of 
bevel gear teeth upon a flat surface, because a bevel 
gear is essentially spherical in its nature ; for practical 
purposes we draw a line c A perpendicular to O c, 
letting c A reach the centre line O A, and take c A 
as the radius of a circle upon which to lay out the 
teeth. This is shown at c n m, Fig. 22. For con- 
venience the line c A is sometimes called the back 
cone radius. 

Let us take, for an example, a bevel gear and a Fi Ex .4 mi>le ' 
pinion 24 and 18 teeth, 5 pitch, shafts at right angles. 
To obtain the forms of the teeth and the data for 
cutting, we need to draw a section of only a half of 
each gear, as in Fig. 22. 



42 BROWN & SHAItPE MFG. CO., 

1. Draw the centre lines A and B 0. then the 
lines g h and c d. and the gear blank lines as des- 
cribed in Chapter IX. Extend the lines op' and o p 
until they meet the centre lines at A B' and A B. 

2. With the radius A c draw the arc c n m, which 
we take as the geometrical pitch circle upon which to 
lay out the teeth at the large end. The distance Ac 
is taken as the radius of the geometrical pitch circle 
at the small end : to avoid confusion an arc of this 
circle is drawn at c n m' about A. 

3. For the pinion we have the radius B c for the 
geometrical pitch circle at the large end and Be for 
the small end: the distance B c is transferred to 
B c . 

4. Upon the arc cum lay off spaces equal to the 
tooth thickness at the large pitch circle, which in our 
example is .314'. Draw the outlines of the teeth as 
in previous chapters : for single curve teeth we draw a 
semi-circle upon the radius A c. and proceed as des- 
cribed in chapter III. For all bevel gears that arc to 
be cut with a rotary disk cutter, or a common gear 
cutter, single curve teeth are chosen : and no attempt 
should be made to cut double curve teeth. Double 
curve teeth can be drawn by the directions given in 
chapters VII aud VIII. We now have the form of 
the teeth at the large end of the gear. Eepeat this 
operation with the radius B C about B, and we have 
the form of the teeth at the large end of the pinion. 

5. The tooth parts at the small end are designated 
by the same letters as at the large, with the addition 
of an accent mark to each letter, as in the right hand 
column, Fig. 22, the clearance, f. however, is usually 
the same at the small end as at the large, for con- 
venience in cutting the teeth. 

Sizes of the The sizes of the tooth parts at the small end are in 
tooth parts. 

the same proportion to those at the large end as 

the line c is to c. In our example O c is _ . 

and O c is 3" : dividing c' by c we have -f. or 

.666, as the ratio of the sizes at the small end to those 



PROVIDENCE, R. J. 



43 





CO 





O 




u 




IE 




o 


JH 


h- 


e* 


Q. 




Lu 


fe( 


o 




UJ 




£E 




z> 




h- 




<c 




> 




DC 




ID 




(J 



44 BROWN A- SHARFE MFG. CO. 

at the large : t is .209 :>rf of .314". and so on. If 
the distance n m is equal to the outer tooth thickness, 
t, upon the arc c n ni, the lines n A and m A will be a 
distance apart equal to the inner tooth thickness t 
upon the arc c n m . The addendum, s . and the 
working depth. D . are at o c and o p . 

6. Upon the arcs c n m and c '" we draw the forms 
of the teeth of the gear and pinion at the inside. 
Example of As an example of the cutting; of bevel sears with 

Cuttins. -,• i , 

rotary disk cutters, or common gear cutters, let us 
take a pair of 8 pitch. 1'2 and 24 teeth, shown in 
Fig. 23. 

Length of In making the drawing it is well to remember that 
nothing is gained by having the face F E longer than 
five times the thickness of the teeth at the large 
pitch circle, and that even this is too long when it is 
more than a third of the apex distance c. To cut a 
bevel gear with a rotary cutter, as in Fig. 24, is at 
best but a compromise, because the teeth change pitch 
from end to end. so that the cutter, being of the right 
form for the large ends of the teeth can not be right 
for the small ends, and the variation is too great when 
the length of face is greater than a third of the apex 
distance c. Fig. 23. In the example, one-third of 
the apex distance is T V . but F E is drawn only a 
half inch, which even though rather short, has changed 
the pitch from 8 at the outside to finer than 11 at the 
inside. Frequently the teeth have to be rounded over 
at the small ends by filing : the longer the teeth the 
more we have to file. If there is any doubt about the 
strength of the teeth, it is better to lengthen at the 
large end. and make the pitch coarser rather than to 
lengthen at the small end. 

Data : : These data are needed before beginning to cut : 

1. The pitch and the numbers of the teeth the same 
as for spur gears. 

2. The data for the cutter, as to its form : some- 
times two cutters are needed for a pah' of bevel gears. 

3. The whole depth of the tooth spaces, both at 



:-:Ti^r. 



PEOVIDENCE, E. I. 



45 




S = .200" 

D"= .400" 

S+/= .231" 

D"+/ = .431" 



D'"= .266" 
S'+f =.165" 
D'"-h/ = .298" 



Fig. 22. 

BEVEL GEARS, FORM AND SIZE OF TEETH, 



46 BROWN & SHARPE MFG. CO. 

the outside and inside ends ; D" + f at the outside, 
and D'" -J- f at the inside. 

4. The thickness of the teeth at the outside and at 
the inside ; t and t'. 

5. The height of the teeth above the pitch lines at 
the outside and inside ; s and s'. 

6. The cutting angles, or the angles that the path 
of the cutter makes with the axes of the gears. In 
Fig. 23 the cutting angle for the gear c D is A Op, 
and the cutting angle for the pinion is B O o. 

selection of The form of the teeth in one of these gears differs 

cutters. 

so much from that in the other gear that two cutters 
are required. In determining these cutters' we do not 
have to develop the forms of the gear teeth as in 
Fig. 22 ; we need merely measure the lines A c and 
B c, Fig. 23, and calculate the cutter forms as if these 
distances were the radii of the pitch circles of the 
gears to be cut. Twice the length Ac, in inches, 
multiplied by the diametral pitch, equals the number 
of teeth for which to select a cutter tor the twenty- 
four-tooth gear : this number is about 54, which calls 
for a number three bevel gear cutter in the list of 
bevel gear cutters, page 61. Twice B c, multiplied 
by 8, equals about 13, which indicates a No. 8 bevel 
gear cutter for the pinion. This method of selecting 
cutters is based upon the idea of shaping the teeth as 
nearly right as practicable at the large end, and then 
filing the small ends where the cutter has not rounded 
them over enough. 

In Fig. 25 the tooth L has been cut to thickness at 
both the outer and inner pitch lines, but it must still 
be rounded at the inner end. The teeth M M have 
been filed. In thus rounding the teeth ihey should not 
be filed thinner at the pitch lines. 

There are several things that affect the shape of the 
teeth, so that the choice of cotters is not always so 
simple a matter as the taking of the lines A c and 
B c as radii. 

In cutting a bevel gear, in the ordinary gear cutting 



PROVIDENCE, R. I. 



47 




BEVEL GEAR DIAGRAM FOR DIMENSIONS. 



4* BROWS i SHARPE 3JTG. CO. 

i_ii:_ines. the finished spaces are not always of the 
same form as the cutter might be expected to makr. 
because of the changes in the positions of the eutter 
and of the gear blank in order to cut the teeth of the 
right thickness at both ends. The cutter must of 
course be thin enough to pass through the small end of 
the spaces, so that the large end has to be cut to the 
right width by adjusting either the cutter or the blank 
side wise, then rotating the blank and cutting twice 
around. 

Widening T-T.15. Ill r \Z- '--^ ■ i 1'fiT :V_ 11 u C~I"~ _ ;._r 5-rt Z~. _ V r 

the Urge a space widened at the large end e . and the last chip 
to be cut off by the right side of the eutter, the cutter 
having been moved to the left, and the blank rotated 

in the direction of the arrow : in a Universal Milling 

_ 

Machine the same result would be attained by moving 
the blank to the right and rotating it in the direction 
of the arrow. It may be well to remember that in 
setting to finish the side of a tooth, the tooth and the 
cutter are first separated side wise, and the blank is 
then rotated by indexing the spindle to bring the large 
Tteeth Mr _ end of the tooth up against the cutter. This trnds 
't^re "JJ^jj not only to cut the spaces wider at the large pitch 
at root. circle, but also to cut off still more at .the face of the 

tooth ; that is, the teeth may be cut rather thin at the 
face and left rather thick at the root. This tendency 
is greater as a cutting angle BOo. Fig. 23, is smaller, 
or as a bevel gear approaches a spur gear, because 
when the cutting angle is small the blank must be 
rotated through a greater arc in order to set to cut the 
right thickness at the outer pitch circle. This can be 
understood by Figs. 26 and 27. Fig. i-i is a radial- 
toothed clutch, which for our present purpose can be 
regarded as one extreme of a bevel gear in which the 
trrth are cut square with the axis: the dotted lines 
indicate the different positions of the cutter, the side 
of a tooth being finished by the side of the cutter that 
is on the centre line. In setting to cut these : 
there is the same side adjustment and rotation of the 



PROVIDENCE, R. I. 




Tig. 24 



SETTING BEVEL GEAR CUTTER 
OUT OF CENTRE. 




50 



BROWN & SHAHl'K MFG. CO. 



spindle as in a bevel gear, but there is no tendency to 
make a tooth thinner at the face than at the root. On 
the other hand, if we apply these same adjustments to 
a spur gear and cutter, Fig. 27, we shall cut the face 
F much thinner without materially changing the thick- 
ness of the root R. 





Fig* 26 

Almost all bevel gears are between the two extremes 
of Figs. 26 and 27, so that when the cuttiug angle 
B O o, Fig. 23, is smaller than about 30°, this change 
in the form of the spaces caused by the rotation of the 
blank maybe so great as to necessitate the substitution 




Fig 1 28 

FINISHED GEAR. 



PROVIDENCE, R. I. 51 

of a cutter that is narrower at e e', Fig. 24, than is 
called for by the way of figuring that we have just 
given : thus in our own gear cutting department we 
might cut the pinion with a No. G cutter, instead of a 
No. 8. The No. 6, being for 17 to 20 teeth, cuts the 
tooth sides with a longer radius of curvature than the 
No 8, which may necessitate considerable filing at the 
small ends of the teeth in order to round them over 
enough. Fig. 28 shows the same gear as Fig. 25, but 
in this case the teeth have all been filed similar to 
M M, Fig. 25. 

Different workmen prefer different ways to com- Filing the 

,i ,,. £\-i Trri teeth at the 

promise m the cutting or a bevel gear. When a email end. 

blank is rotated in adjusting to finish the large end of 

the teeth there need not be much filing of the small 

end, if the cutter is right, for a pitch circle of the 

radius B c, Fig. 23, which for our example is a No. 8 

cutter, but the tooth faces may be rather thin at the 

large ends. This compromise is preferred by nearly 

all workmen, because it does not require much filing 

of the teeth : it is the same as is in our catalogue by 

which we fill any order for bevel gear cutters, unless 

otherwise specified. This means that we should send c 2SS? ti whcn 

a No. 8, 8-pitch bevel gear cutter in reply to an order J ee j*j are to 

for a cutter to cut the 12-tooth pinion, Fig. 23 ; while 

in our own gear cutting department we might cut the 

same pinion with a No. 6, 8-pitch cutter, because we 

prefer to file the teeth at the small end after cutting 

them to the right thickness at the faces of the large 

end. We should take a No. G instead of a No. 8 only 

for a 12-tooth pinion that is to run with a gear two or 

three times as large. We generally step off to the 

next cutter for pinions fewer than twenty-five teeth, 

when the number for the teeth has a fraction nearly 

reaching the range of the next cutter : thus, if twice 

the line B c in inches, Fig. 23, multiplied by the 

diametral pitch, equals 20.9, we should use a No. 5 

cutter, which is for 21 to 25 teeth inclusive. In 

filling an order for a gear cutter, we do not consider 



52 



x a- shap.fz :;: 



the fraction but send the cotter indicated by the whole 
number. 

Later on ^c- will refer to other compromises that are 
madv :: :lr sotting >f bevel gears. 

The sizes of the H-piteh tooth parts, Fig. 23. at the 
large end. are copied from the table of spar _ 
teeth, pages 145 to 148. 

The distance Oc is seven-tenths of the apex 
tance Oc. so that the sizes of the tooth y ts t the 
gear cutting small end, rxieptf, are seven-tenths the larse. The 
order for cutting these gears goes to the workmen in 
tV-is :o::n : 



Form of 



Senir _ : ~ 

ir. ;. ::.:re 



Lab 5b i.tear. 








P = 5 








N = 24 








y — f = .270' D — 


f 


= 


.195' 


t = .: 


t 


= 


.137' 


s = .12o 


5 


= 


. ^7 



Cot: ins: Ansle = 59° 10' 



- r 



Small Gkar. 



X 



C:::::: Angle = l. : IS 

Fig. B2 is aside view of a Geai Cutting Machine. 
A evel gar blank A is held by the index spindle B. 
The cotter C is sanied y the cutter-slide D. The 
cutter-slide- carriag E n be set to the cutting angle, 
the degrees being indicated on the quadrant F . 

Fig l Z is a plan of the machine : in this view the 
: ::er-slide-carriage. in order to show the detaiis 
little p'ainer, is not set to an angle. 

Before bes^nnino; to cut the cutter is set central with 
the index spindle and the dial G is set :; zero, so 
that we can adjust the softer to any required distance 
out of centre, in either direction. Set the c itternBlide- 
carriage E. Fig. 32, to the cutting angle of the gear. 
which for 24-teeth is 59° 10': the quadrant beii _ 
divided to half -degrees, vre estimate thai l( or \ de- 



PROVIDKNCK, K. I 



53 



gree more than 59°. Mark the depth of the cut at the 
outside, as in Fig. 30 : it is also well enough to mark 
the depth at 'he inside as a check. The thickness of 
the teeth at the large end is conveniently deter- 
mined by the solid gauge, Fig. 29. The gear-tooth 




J GEAR TOOTH GAUGE. 



DEPTH 
GAUGE. 



_Fi#. 30 



GEAR TOOTH CALIPER. 
JFig.31 



vernier caliper, Fig. 31, will measure the thickness of 
teeth up to 2 diametral pitch. In the absence of the 
vernier caliper we can file a gauge, similar to Fig 29, 
to the thickness of the teeth at the small end. 

The index having been set to divide to the right 8 i^.°o? tooth 
number we cut two spaces central with the blank, being finished 
leaving a tooth between that is a little too thick, as in 
the upper part of Fig. 25. If the gear is of cast iron, 
and the pitch is not coarser than about 5 diametral, 
this is as far as we go with the central cuts, and we 
proceed to set the cutter and the blank to finish first 
one side of the teeth and then the other, going around 
only twice. The tooth has to be cut away more in 
proportion from the large than from the small end, 
which is the reason for setting the cutter out of centre, 
as in Fig. 24. 



54 



BllOWN & SHARPE MFG. CO. 




Fig. 32 



AUTOMATIC GEAR CUTTING MACHINE. 



SIDE ELEVATION. 



rrsoviDENCE, r. i. 55 

It is important to remember that the part of the 
cutter that is finishing one side of a tooth at the pitch 
line should be central with the gear blank, in order to 
know at once in which direction to set the cutter out of 
centre. We can not readily tell how much out of 
centre to set the cutter until we have cut and tried, 
because the same part of a cutter does not cut to the 
pitch line at both ends of a tooth. As a trial distance 
out of centre we can take about one-tenth to one- 
eighth of the thickness of the teeth at the large end. 
The actual distance out of centre for the 12-tooth 
pinion is .021" : for the 24-tooth gear, .030", when 
using cutters listed in our catalogue. 

After a little practice a workman can set his blank Necessity of 

1 central cuts. 

the trial distance out of centre, and take his first cuts, 
without any central cuts at all ; but it is safer to take 
central cuts like the upper ones in Fig. 25. The 
depth of cut is partly controlled by the index-spindle 
raising-dial-shaft H, Fig. 33, which determines the 
height of the index spindle, and partly by the position 
of the cutter spindle. We now set the cutter out of 
centre the trial distance by means of the cutter-spindle 
dial-shaft, I, Fig. 33. The trial distance can be about 
one-tenth the thickness of the tooth at the large end 
in a 12-tooth pinion, and from that to one-eighth the 
thickness in a 24-tooth gear and larger. The principle 
of trimming the teeth more at the large end than at 
the small is illustrated in Fig. 24, which is to move 
the cutter away from the tooth to be trimmed, and 
then to bring the tooth up against the cutter by 
rotating the blank in the direction of the arrow. Adustments 

The rotative adjustment of the index spindle is 
accomplished by loosening the connection between the 
index worm and the index drive, and turning the worm,: 
the connection is then fastened again. The cutter is 
now set the same distance out of centre in the other 
direction, the index spindle is adjusted to trim the 
other side of the tooth until one end is down nearly 
to the right thickness. If now the thickness of the 



D6 BROWN & SHAKPE MFG. CO. 

small end is in the same proportion to the large end as 
Oc' is to Oc, Fig. 23, we can at once adjust to trim 
the tooth to the right thickness. But if we find that 
the large end is still going to be too thick when the 
small end is right, the out of centre must be increased. 
It is well to remember this : too much out of centre 
leaves the small end proportionally too thick, and too 
little out of centre leaves the small end too thin. 

After the proper distance out of centre has been 
learned the teeth can be finish-cut by going around out 
of centre first on one side and then on the other with- 
out cutting any central spaces at all. The cutter 
spindle stops, J J, can now be set to control the out 
of centre of the cutter, without having to adjust by 
the dial G. If, however, a cast iron gear is 5-pitch 
or coarser it is usually well to cut central spaces first 
and then take the two out-of -centre cuts, going around 
three times in all. Steel gears should be cut three 
times around. 

Blanks are not always turned nearly enough alike to 
be cut without a different setting for different blanks. 
If the hubs vary in length the position of the cutter 
spindle has to be varied. In thus varying, the same 
depth of cut or the exact J)" -f" f may not always be 
reached. A slight difference in the depth is not so 
objectionable as the incorrect tooth thickness i hat it 
may cause. Hence, it is well, after cutting once 
around and finishing one side of the teeth, to give 
careful attention to the rotative adjustment of the 
index spindle so as to cut the right thickness. 

After a gear is cut, and before the teeth are filed, it 
is not always a very satisfactory-looking piece of work. 
In Fig. 25 the tooth L is as the cutter left it, and is 
ready to be filed to the shipe of the teeth M M, which 
have been filed. Fig. 34 is the pair of gears that we 
have been cutting ; the teeth of the 12-tooth pinion 
have been filed. 



PROVIDENCE, K. I. 



57 




58 



BROWN" & SHARPE MFG. CO. 



a second ^ second approximation in cutting with a rotary 

approxima- x l ^ 

tion- cutter is to widen the spaces atthe large end by swing- 

ing either the index spindle or the cutter-slide-carriage, 
so as to pass the cutter through on an angle with the 
blank side-ways, called the side-angle, and not rotate 
the blank at all to widen the spaces. This side-angle 
method is employed in our No. 11 Automatic Bevel 
Gear Cutting Machines : it is available in the manufac- 
ture of bevel gears in large quantities, because with 
the proper relative thickness of cutter, the tooth- 
thickness comes right by merely adjusting for the 
side-angle ; but for cutting a few gears it is not much 
liked by workmen, because, in adjusting for the side- 
angle, the central setting of the cutter is usually lost, 
and has to be found by guiding into the central slot 
already cut. if the side-angle mechanism pivots about 
a line that passes very near the small end of the tooth 
to be cut, the central setting of the cutter may not 
be lost. In widening the spaces at the large end, 
the teeth are narrowed practically the same amount at 
the root as at the face, so that this side-angle method 
requires a wider cutter at e e . Fig. 24. than the first, 
or rotative method. The amount of filing required 
to correct the form of the teeth at the small end is 
about the same as in the first method. 

a third ap- A third approximate method consists in cutting 

proximation. . . 

the teeth right at the large end by going around at 
least twice, and then to trim the teeth at the small end 
and toward the large with another cutter, going around 
at least four times in all. This method requires skill 
and is necessarily a little slow, but it contains possi- 
bilities for considerable accuracy. 
a fourth ap- A fourth method is to have a cutter fully as thick as 

proximation. , . 

the spaces at the small end, cut rather deeper than 
the regular depth at the large end, and go only once 
around. This is a quick method but more inaccurate 
than the three preceding : it is available in the manu- 
facture of large numbers of gears when the tooth-face 



PROVIDENCE, K. I. 



59 




Fig. 34 



FINISHED GEAR AND PINION 



60 



BROWN & SIIARPE MFG. CO. 



is sho.rt compared with the apex distance. It is little 
liked, and seldom employed in cutting a few gears : it 
may require some experimenting to determine the form 
of cutter. Sometimes the teeth are not cut to the 
regular depth at the small end in order to have them 
thick enough, which may necessitate reducing the 
addendum of the teeth, s, at the small end by turning 
the blank down. This method is extensively employed 
by chuck manufacturers. 

A machine that cuts bevel gears with a reciprocating 
motion and using a tool similar to a planer tool is 
called a Gear Planer and the gears so cut are said to 
be planed. 
pianino- G f ^ ne f° rm of Gear Planer is that in which the prin- 
bevei gears. c [pi e embodied is theoretically correct ; this machine 
originates the tooth curves without a former. Another 
form of the same class of machines is that in which the 
tool is guided by a former. 

Usually the time consumed in planing a bevel gear 
is greater than the time necessary to cut the same gear 
with a rotary cutter, thus proportionately increasing 
the cost. 

Pitches coarser than 4 are more correct and some- 
times less expensive when planed ; it is hardly prac- 
ticable, and certainly not economical, to cut a bevel 
ge-ir as coarse as 3P. with a rotary cutter. In gears as 
fine as 1GP. planing affords no practical gain in quality. 

While planing is theoretically correct, yet the wear- 
ing of the tool may cause more variation in the thick- 
ness of the teeth than the wearing of a rotary cutter, 
and even a planed gear is sometimes improved by filing. 
Mounting of ^ gears are not correctly mounted in the place where 
gears. they are to run, they might as well not be planed. In 

fact, after taking pains in the cutting of any gear, 
when we come to the mounting of it we should keep 
right on taking pains. 
Angles and The method of obtaining the sizes and angles per- 
geais. taining to bevel gears by measuring a drawing is quite 

convenient, and with care is fairly accurate. Its 



PROVIDEXCK, K. I. 



61 



accuracy depends, of course, upon the careful measur- 
ing of a good drawing. We may say, in general, that 
in measuring a diagram, while we can hardly obtain 
data mathematically exact, Ave are not likely to make 
wild mistakes. Some years ago we depended almost 
entirely upon measuring, but siuce the publication of 
this "Treatise" and our " Formulas in Gearing" we 
calculate the data without any measuring of a drawing. 
In the ""Formulas in Gearing " there are also tables 
pertaining to bevel gears. 

Several of the cuts and some of the matter in this 
chapter are taken from an article by O. J. Beale, in 
the "American Machinist," June 20, 1^95. 

Cutters for Mitre and Bevel 
Gears. 



Diametral 


Diameter of 


Hole in 


Pitch. 


Cutter. 


Cutter. 


4 


8 3-8" 


1 1-4" 


5 


3 1-16 


1 1 


6 


2 3-4 


1 1-16 


8 


2 1-2 


1 1 


10 


2 1-8 


7-8 


12 


2 


ii 


14 


2 


(. i 


16 


1 15-10 


1 1 


20 


1 7-8 


I c 


24 


1 3-4 


( . 




WORM WHEEL. 



Number of Teeth, 34. Circular Pitch, 2J/.j. 

Throat Diameter, 44.39". Outside Diameter, 46' 



C3 



CHAPTER XI. 
WORM WHEELS— SIZING BLANKS OF 32 TEETH AND OVER. 



A worm is a screw made to mesli with the teeth of Worm. 
a wheel called a worm-wheel. As implied at the end of 
Chapter IV., a section of a worm through its axis is, in 
outline, the same as a rack of corresponding pitch. 
This outline can be made either to mesh with single or 
double curve gear teeth ; but worms ' are usually made 
for single curve, because, the sides of involute rack 
teeth being straight (see Chapter IV.), the tool for 
cutting worm-thread is more easily made. The thread- 
tool is not usually rounded for giving fillets at bottom 
of worm-thread. 

The axis of a worm is usually at rigl t singles to the 
axis of a worm wheel: no other angle of axis is treated 
of in this book. 

The rules for circular pitch apply in the size of tooth 
parts and diameter of pitch-circle of worm-wheel. 

The pitch of a worm or screw is sometimes given in Pitch of worm 
a way different from the pitch of a gear, viz. : in num- 
ber of threads to one inch of the length of the worm or 
screw. Thus, to say a worm is 2 pitch may mean 2 
threads to the inch, or that the worm makes two turns 
to advance the thread one inch. But a worm may be 
double-threaded, triple-threaded, and so on; hence 
to avoid misunderstanding, it is better always to call 
the advance of the worm thread the lead. Thus, a worm-Thread 
worm-thread that advances one inch in one turn we 
call one-inch lead in one turn. A single-thread worm 
4 to 1" is J" lead. We apply the term pitch, that is 
the circular pitch, to the actual distance between the 
threads or teeth, as in previous chapters. In single- 
thread worms the lead and the pitch are alike. If we 
have to make a worm and wheel so many threads to 



04 



BROWN & SHAEPE MFG. CO. 




FIG. 35— WORM AND WORM-WHEEL 

The thread of Worm is left-handed ; Worm is single-threaded 



PROVIDENCE, R. I. 



05 




: -~ 



: : 



ir inehj we irs: ziride 1" hw lie number of threads Mm 
ne imdk 9 and the quotient iM-ihe circular ptidL Hence, 

~_r-"-f:-" - F .z. : is - ::.;-".:::.::_. X :'/'j:::. 
sxprcoBBa tI .:.::- — bat i3 mean: f : ar pitch. 
~ : T. r:::l « :'_e :>. .: rf :: :^ nz . ei :: use. 

: : ri.-.:r ~:f: t " : m= i I : . :k=. T _ e 

_~m.ljer of threads to one iceh linear, is the reciprocal 
: i ~ . r . . t i - tie: . . 

Multiply 3.1416 bj the number of threads to one 
inch, and the product will be the diametral pitch of the 
— ::=:- irr. r_~j. -r=i:~"-i =.".".: ^ iir'.'.r-:*-: f..:r 3 
worm advancing 1" in 1^ turns that: 
Dnvimg of Iiead=§" or .7-5 . Iane.xT ::::': ::P = t " r 

-i^^^-AT " - - ii-=-~-il; :: :_ :: F = S : "" ^rr :e.":*_-e- :::; : :_ • -- 

Z: _ :t "_E-~_zr :_ ~::z_ ezlI — _-ee. ~r . ': : . __ 
V ::■. . =: ._ :_ : :i '. :::n 

1 iTi- :-:_:-: jzlt A J iz.1 u\:_ i: - - :± :_e- 
_:- t: ii_ :: ~_t ELLim-rT-EEr :: •'::;_-:_;_ - 

' _ t. ._ -: ".t : : -L-e-^t i — : \ : :_;- Li; :z :_t ii=- 
- .-. _ _ t _ : : : _ t - - ~ L ;. :" ifzi 1 : i_ = : - 

3. From lay :± the itmtwocr c O equal to the 
riiii= :: :_- — ::_ T_t iiizi-E-Tr^ :: i — :-_ is _ ri.- 

• — 

-i L^~ :z" :1t ii-:^z:»5-~ : ; evlI ~ - -e ~ . :_ e~: . : 
: _ - _ t - ! - t ;-. : _ : e- ;- : :::_::::::_-;■:: t 

:" I_ ::__ . .\-.~. - i_:,- eh:! 5 :: j Xlr-s* 

r-e~:'TETz: tie ~_:1- Iie^-tEe: :: ~::_ :-._.*. :_r : _- 
.:": ;: ': :::_ :: -:::_-:_. e '. 

: I : .■.- ! :_" . :.: :-_ .-.ei_ _t :::!"::- :" ' — :e_ 
A ! r_r-T __-- .-. r - : '.-'_ :: :i:- :: — :l_--_tt.. 

Z_e . :^_ ml : ~_e.^" ".. : - i : : i: '.'. rz.v~ -. _ 
: ; :_: ; 

~_i5 : t.::::_ :- - -" : :_.-:t- :_r :^tln.e 

:: — ::n-— LettI. F:: :. :tt"_ _ ". : e :_- " t_ . _ 

_ _t:t: :: D. ~_:.^1 ::~ :..^>-z. i: :_t :_: : : :r 
5i_:._tE-: L_e.l_t:t. :: — jl-e-t! - . - 7-_ - JS/f---* ; .«'f 



PROVIDENCE, R. I. 



67 



threaded with a tool of the same angle as the tool that 
threads the worm, the end of the tool bein<r .335 of 
the linear pitch ; the hob is then grooved to make teeth 
for cutting, and hardened. 

The whole diameter of hob should be at least 2 /* Proportionsof 

«" Hob. 

or twice the clearance larger than the worm. In our 
relieved hobs the diameter is made about .005" to .010" 
larger to allow for wear. The outer corners of hob-thread 
can be rounded down as far as the clearance distance. 
The width at top of the hob-thread before rounding 
should be .31 of the linear, or circular pitch — .31P'. 
The whole depth of thread is thus the ordinary work- 
ing depth plus the cle:iranee=D"+/. The diameter 
at bottom of hob-thread should he 2/+. 005" to .010" 
larger than the diameter at bottom of worm-thread. 




Fig. 37— HOB. 

For thread-tool and worm-thread see end of Chapter 
IV. 

In the absence of a special worm gear cutting ma - tn H( H r l) t0 use 
chine, the teeth of the wheel are first cut as nearly to the 
finished form as practicable; the hob and worm-wheel 
are mounted upon shafts and hob placed in mesh, it is 
then rotated and dropped deeper into the wheel until the 
teeth are finished. The hob generally drives the worm- 
wheel.during this operation. The Universal Milling Ma- universal 
chine is convenient for doing this work ; with it the d is- chine used in 

Hobbing. 



68 



BROWN & SHARPE MFG. CO. 



mJf 




Fig. 38. 



PROVIDENCE, R. I. 



G9 



^£ H ? c,R s^ 




Fig. 39. 



■ 



::.:":■ & ~z.-.:.:-i i::} ::. 



tan ce be t ween axes of wo rm and wheel ea n be n oted . In 
making wheels in quantities it is be!ter to have a ma- 
chine in which the work spindle is drive a y rearing, 
so that the hob can cut the teeih from the solid with- 
— -va wheel OD * gashing. The object of hobbing a wheel is to _ - 

iiH ivz-rt more bearing surface of the teeth upon worm-thread. 

The worm-wheels. Figs. 35 aud 43. I obbed. 

wonn-wiieei If we make the diameter of a worm-wheel blank, that 

leas than 30 is to have less than 30 teeth, by the common rules 

for sizing blanks, and finish the teeth with a hob. 

shall find the flanks of teeth n sax the bottom to be un- 

inierierenee dercut or hollowing. This is caused bv the interfer- 
ed __Lresi B.~i , : __ r . v 

Flank. ence spoken of m Chapter \ I. Inirty teeth was there 

given as a limit, which will be right when teeth are 
made to circle arcs. With pressure an^ '.- ~'- : . aid 
rack-teeth with usual addendum, thU ::::e::erence of 
rack-teeth with flanks of gear- teeth commences at 31 
teeth (31^ geometrically), and interferes with nearly 
the whole flank in wheel of 12 re:h. 

: - In Fig 38 the blank for worm- wheel of 12 teeth w ■ ; 

sized by the same rule as given for Fig. 36. The wheel 
and worm are sectioned to show shape of teeth at the 
mid-plane of wheel. The flunks of teeth are undercut 
by the hob. The worm-thread does not have a good 
bearing on flanks inside of A. the bearing being that of 
a corner against a surf a e 

^s- »• In Fig 39 the blank for wheel wae sized sc that pitch- 

circle comes midway between outermost part of tee:_ 
■nd innermost point obtained by worm thread. 

This rule for sizing worm-wheel blanks has been in 
use :: - >me ri:ri:. The hob has cut away flanks of 
teeth still more than in Fig. 3S. The pitch circle in 
Kg. 39 is the same diameter as the pitch-circle in Fig. 
38. The same hob was used for both wheels. The 
flanks in this wheel are so much undercut as to mate- 
rially lessen the bearing surface of teeth and worm- 
thread. 
interference In Chapter ~YX the interference of teeth in high- 
numbered g r ears and racks with flanks of 12 teeth vr 
remedied by rounding off the addenda. Although it 
w raid be move systematic to round off the threads ol 
a worm, making them, like rack-teeth, to mesh w::_ 



PROVIDENCE, R. I. 71 

interchangeable gears, yet this has not generally been 
done, because it is easier to make a worm-thread tool 
with straight sides. 

Instead of cutting away the addenda of worm- 
thread, we can avoid the interference with flanks of 
wheels having less than 30 teeth by making wheel 
blanks larger. 

The flanks of wheel in Fig. 40 are not undercut, be- Fig. 40. 
cause the diameter of wheel is so large that there is 
hardly any tooth inside the pitch-circle. The 
pitch-circle in Fig. 40 is the same size as pitch- 
circles in Figs. 38 and 39. This wheel was sized 
by the following rule : Multiply the pitch diameter of Diameter at 

Throat to Avoid 

the wheel by .937, and add to the product four times interference, 
the addendum (4 s) ; the sum will be the diameter for 
the blank at the throat or small part. To get the 
whole diameter, make a sketch with diameter of throat 
to the foregoing rule and measure the sketch. 

It is impractical to hob a wheel of 12 to about 16 or 
18 teeth when blank is sized by this rule, unless the 
wheel is driven by independent mechanism and not by 
the hob. The diameter across the outermost parts of 
teeth, as at A B, is considerably less than the largest 
diameter of wheel before it was bobbed. 

In general it is well to size all blanks, as by page 66 
and Figs. 36 and 38, when the wheels are to be hobbed ; 
of course the cutter should be thin enough to leave 
stock for finishing. The spaces can be cut the full 
depth, the cutter being dropped in. 

When worm-wheels are not hobbed it is better to 
turn blanks like a spur-wheel. Little is gained by g ^J a n J £ h Li J e a 
having wheels curved to fit worm unless teeth are fin- 
ished with a hob. The teeth can be cut in a straight 
path diagonally across face of blank, to fit angle of 
worm-thread, as in Figs. 41 and 44. 

For dividing wheels to gear-cutting engines the Wheels for 
blanks are turned like a spur-wheel and a cutter about Machines. 
T *g-" larger diameter than the worm, is dropped in, as 
in Figs. 42 and 45. 



: : - 5: 




J _ v 





Fn 40 



PROVIDENCE, K. I. 73 

Some mechanics prefer to make dividing wheels in 

two parts, joined in a plane perpendicular to axis, hob 

teeth , then turn one part round upon the other, match 

teeth and fasten parts together in the new position, 

and hob again with a view to eliminate errors. With 

an accurate cutting engine we have found wheels like 

Figs. 42 and 45, not nobbed, every way satisfactory. 

As to the different wheels, Figs. 43, 44 and 45, when Figures 43, 44 

° , . and 45. 

worm is in right position at the start, the life-time 
of Fig. 43, under heavy and continuous work, will be 
the longest. 

Fig. 44 can be run in mesh with a gear or a rack as 
well as with a worm when made within the angular 
limits commonly required. Strictly, neither two gears 
made in this way, nor a gear and a rack would be 
mathematically exact, as they might bear at the sides 
of the gear or at the ends of the teeth only and not in 
the middle. At the start the contact of teeth in this 
wheel upon worm-thread is in points only; yet such 
wheels have been many years successfully used in ele- 
vators. 

Fig. 45 is a neat-looking wheel. In gear cutting 

engines where the workman has occasion to turn the 
work spindle by hand, it is not so rough to take hold 
of as Figs. 43 and 44. The teeth are less liable to in- 
jury than the teeth of Figs. 43 and 44. 



BROWN & SHAUPE MFG. CO. 




Fig. 41. 



"Worm-wheel with teeth cut in a straight path diagonally across face. 
Worm is double-threaded. 



PROVIDENCE, E. I. 



7a 




Fig. 42. 

Worm and Worm- Wheel, for Gear-cutting Engine. 



7G 



BROWN & SHAIirE MFG. CO. 






UaaMjamik--- 



Fig. 43. 



Fig. 44. 



Fig. 45. 



PROVIDENCE, R. I. 77 

Some designers prefer to take off the outermost part 
of teeth in wheels (Figs. 35 and 43), as shown in these 
two figures, and not leave them sharp, as in Figs. 36 
and 80. 

We do not know that this serves any purpose except 
a matter of looks. 

In ordering worms and worm wheels the centre dis- 
tances should be given. 

If there can be any limit allowed in the centre distance 
it should be so stated. 

For instance, the distance from the centre of a worm 
to the centre of a worm wheel might be calculated at 
6" but 5 31-32" or 6 1-32" might answer. 

By stating all the limits that can be allowed , there 
may be a saving in the cost of work because time need 
not be wasted in trying to make work within narrower 
limits than are necessary. 



HOBS WITH RELIEVED TEETH. 

We are prepared to make hobs of any size with the 
teeth relieved the same as our gear cutters. The teeth 
can be ground on their faces without changing their 
form. The hobs are made with a precision screw so 
that the pitch of the thread is accurate before hard- 
ening. 



BIIOTVX & SHARFE MFG. CO. 




GASHING TEETH OF HOB, 



lO Inches Ou.tside Diameter. 



PROVIDENCE, E. I, 



CHAPTER XII. 

SIZING GEARS WHEN THE DISTANCE BETWEEN CENTRES AND THE 
RATIOS OF SPEEDS ARE FIXED— GENERAL REMARKS— WIDTH 
OF FACE OF SPOR GEARS— SPEED OF GEAR CUTTERS— TABLE 
OF TOOTH PARTS. 



Let us suppose that we have two shafts 14" apart, 
center to center, and wish to connect them by gears so, center dis- 

' J ° tance and Ratio 

that they will have speed ratio 6 to 1. We add the 6 fixed - 
and 1 together, and divide 14" by the sum and get 2" 
for a quotient; this 2", multiplied by 6, gives us the 
radius of pitch circle of large wheel === 12". In the same 
manner we get 2" as radius of pitch circle of small wheel. 
Doubling the radius of each gear, we obtain 24" and 4" 
as the pitch diameters of the two wheels. The two num- 
bers that form a ratio are called the tsrms of the ratio. 
We have now the rule for obtaining pitch-circle diame- 
ter of two wheels of a given ratio to connect shafts a 
given distance apart: 

Divide the center distance bii the sum of the terms of Rule f °r r>u 

° ^ J ameter of Pitch 

the ratio; find the product of tin ice the quotient by each circles. 
term sparately, and the two products ivill he the pitch 
diameters of the two ivheels. 

It is well to give special attention to learning the 
rules for sizing blanks and teeth ; these are much 
oftener needed than the method of forming tooth out- 
lines. 

A blank 1-J-" diameter is to have 16 teeth: what will 
the pitch be? What will be the diameter of the pitch 
circle ? See Chapter V. 

A good practice will be to compute a table of tooth 
parts. The work can be compared with the tables 
pages 86-89. 



Sj BBovrs- jc szat.iz mtg. 

In compute:, i: is well to take re to more than four 
rlaces. ,t to nine places = 3.141592653. •= to nine 
places = .318309886. 

There is no such thing as pure rolling contact in 
teeth of wheels: they always rub, and. in time, will 
wear themselves out of snape and niay become noisy. 

Bevel gears, vriier. correctly ft-raieiL ran smoother 
than spur gears of same diameter and pitch, because 
the teeth continue in contact longer than the teeth of 
spur gears. F r this reas : n annular gears run smoother 
than either bevel or spur gears. 

Sometimes gears have to be cut a little deeper than 
designed, in order to run easily on their shafts. If 
any departure is made in ratio of pitch diameters it :- 
better to have the driving gear the larger, that is, cut 
the follower smaller. For wheels coarser than eight 
diametral pitch (8 P'». it is generally better to cottwi x 
around, when accurate wxk is wanted, also for lar _ -e 
wheels, as the expansion of parts from heat often causes 
inaccurate work when cut but once around. There is 
not so much trouble from heat in plain or we _ e is as 
in arm gears. 
w-. .-_:-; „r Tiie -~: "t"_ c : cast-ii:n _ear :a:es ::•: r/ererai rtir- 
poses can be made to the following rule : 

It v:\J.-i S :y ' -to the 

quo* it : sum toiU be width of face for the pitch 

Example : "VThat width of face for gear 4 P Divid- 
ing 8 by -i and adding J" we obtain 2^", for width of 
face. For change gears on lathes, where it is desira- 
ble not to have face very wide, the following rule can 
be used: 

Divide -± ' -ch and add J 

Bv the latter rule a 4 P change gear would have but 
H race 
Speedof Gear The speed of gear cutters is subject t : s :> many con- 
ditions that dehnite rules cannot be given. TVe append 

table of average speeds. A coarse pitch cutter for 
pinion. 12 teeth, would usually be fed slower than a 
cutter for a large gear of same pitch. 



PROVIDENCE, E. I. 



81 



TABLE OF AVERAGE SPEEDS FOR GEAR-CUTTERS. 



A 




bfi 


tI) S 


». 


u • 




c 


; O 


«w 




^.S ° . 


C3^ A 


S3 « ^"p 




o 


S| 


O 

5S 


ftp g 


w ~ m ~ 
P.3 ^ o 


at § 




m d d 

Kg 


t- d *■" ■— c 

O.SM S3 
&S3^£ 


Diametral 
of Cut 


u 

5 


aa °H 

P S3 -u 

JJ3 3 


Er = ° ^ 


3cm +» 
7J CO 

r - 


2<^ 9^ 


^s5 


r- "S — OQ 


2 


5 in. 


24 


18 


.025 in 


.011 in. 


.60 in. 


. 20 in. 


2* 


±i " 


30 


24 


.028 " 


.013 " 


.84 " 


.31 " 


3 


16 


36 


28 


.031 " 


.015 " 


1.12 " 


.42 " 


4 


3f " 


42 


32 


.034 " 


.017 " 


1.43 « 


.54 " 


5 


3 T V " 


50 


40 


.037 " 


.019 " 


1.85 " 


.76 " 


6 


2U- " 

16 


75 


55 


.030 " 


.016 " 


2.25 " 


.88 " 


7 


2-9- '« 

"" 1 6 


85 


65 


.032 " 


.018 " 


2'. 72 « 


1.17 " 


8 


9± " 

~2 


95 


75 


.034 " 


020 " 


3.23 " 


1.50 " 


10 


01 « 

Z S 


125 


90 


.026 " 


.014 " 


3.25 " 


1.26 " 


12 


2 " 


135 


100 


.027 " 


.017 " 


3.64 " 


1.70 " 


20 


H " 


145 


115 


.029 " 


.021 " 


4,20 " 


2.41 " 


32 


H « 


160 


135 


.031 '• 


.025 " 


4.96 " 


3.37 " 



In brass the speed of gear-cutters can be twice as B ^ g e e d s ? 
fast as in cast iron. Clock-makers and those making a 
specialty of brass gears exceed this rate even. A 12 P 
cutter has been run 1,200 (twelve hundred) turns a 
minute in bronze. A 32 P cutter has been run 7,000 
(seven thousand) turns a minute in soft brass. 

In cutting 5 P cast-iron gears, 75 teeth, a No. 1, BP^^g 9 
cutter was run 136 (one hundred and thirty-six) turns 
a minute, roughing the spaces out the full 5 P depth ; 
the teeth were then finished with a 5 P cutter, running 
208 (two hundred and eight) turns a minute, feeding 
by hand. The cutter stood well, but, of course, the 
cast iron was quite soft. A 4 P cutter has finished 
teeth at one cut, in cast-iron gears, 86 teeth, running 48 
(forty-eight) turns a minute and feeding T y at one 
turn, or 3 in. in a minute. 

Hence, while it is generally safe to run cutters as in 
the table, yet when many gears are to be cut it is well to 
see if cutters will stand a higher speed and more feed. 

In gears coarser than 3 P it is more economical to 
cut first the full depth with a stocking cutter and then 
finish with a gear cutter. This stocking cutter is made 



82 BROWN & SHAEPE MEG. CO. 

on the jDrinciple of a circular splitting saw for wood. 
The teeth, however, are not set ; but side relief is ob- 
tained by niakino- sides of cutter blank hollowing. The 
shape of stocking cutter can be same as bottom of 
spaces in a 1 --tooth gear, and the thickness of cutter 
can be J- of the circular pitch, see page 40. 
Keep cutters The matter of keeping cutters sharp is so important 
that it has sometimes been found best to have the work- 
man grind them at stated times, and not wait until he 
can see that the cutters are dull. Thus, have him 
m.ind everv two hours or after cutting a stated number 
of gears. Cutters of the style that can be ground 
upon their tooth faces without changing form are rap- 
idly destroyed if allowed to run after they are dull. 
Cutters are oftener wasted bv trying to cut with them 
Avhen they are dull than by too much grinding. Grind 
Ihe faces radial with a free cutting wheel. Do not let 
the wheel become glazed, as this will draw the temper 
of the cutter. 

In Chapter YI. was given a series of cutters for cut- 
ting gears having 12 teeth and more. Thus, it was 
there implied that any gear cf same pitch, having 135 
teeth, 13G teeth, and so on up 1 :> the largest gears, and, 
also, a rack, could be cut with one cutter. If this cut- 
ter is 4 P, we would cut with it all 4 P gears, having 
135 teeth or more, and we would also cut with it a 4P 
rack. Now. instead of alwavs ref erring to a cutter bv 
the number of teeth in gears it is designed to cut, it 
has been found convenient to designate it by a letter 
or by a number. Thus, we call a cutter of 4 P, made 
to cut gears 135 teeth to a rack, inclusive, Nj. 1, 4 P. 

We have adopted numbers for designating involute 
involute Gear orear-cutters a-s in the following table: 

Cutters. 

No. 1 will cut wheels from 135 teeth to a rack inclusive. 



55 


CI 


134 teeth 


35 


" 


54 " 


26 


u 


34 - : 


21 


« 


25 ki 


17 


.. 


20 - 


14 


it 


16 " 


12 


« 


13 " 



PROVIDENCE, R. I. 83 

By this plan it takes eight cutters to cut all gears 
having twelve teeth and over, of any one pitch. 

Thus it takes ei»ht cutters to cut all involute 4 P 

O 

gears having twelve teeth and more. It takes eight 
other cutters to cut all involute gears of 5 P, having 
12 teeth and more. A No. 8, 5 P cutter cuts only 5 P 
gears having 12 and 13 teeth. A No. 6, 10 P cutter 
cuts only 10 P gears having 17, 18, 19 and 20 teeth. 
On each cutter is stamped the number of teeth at the 
limits of its range, as well as the number of the cutter. 
The number of the cutter relates only to the number 
of teeth in gears that the cutter is made for. 

In ordering cutters for involute spur-gears two things 
must be given : 

1. Either the number of teeth to be cut in the qear How to order 

J J Involute Cut- 

or the number of the cutter, as given in the foregoing ters. 
table. 

2. Either the pitch of the gear or the diameter and 
number of teeth to be cut in the gear. 

If 25 teeth are to be cut in a 6 P involute gear, the 
cutter will be No. 5, 6 P, which cuts all 6 P gears from 
21 to 25 teeth inclusive. If it is desired to cut gears 
from 15 to 25 teeth, three cutters will be needed, No. 
5, No. 6 and No. 7 of the pitch required. If the pitch 
is 8 and gears 15 to 25 teeth are to be cut, the cutters 
should be No. 5, 8 P, No. 6, 8 P, and No. 7, 8 P. 

For each pitch of epicycloidal, or double-curve gears, Epicycioidai 
24 cutters are made. In coarse-pitch gears, the varia- cur ^ e cutters, 
tion in the shape of spaces between gears of consecu- 
tive-numbered teeth is greater than in fine-pitch gears. 
A set of cutters for each pitch, to consist of so large a 
number as 24, has been established because double curve 
teeth have generally been preferred in coarse-pitch gears, 
though the tendency of late years is toward the involute 
form. 

Our double curve cutters have a guide shoulder on each 
side for the depth to cut. When this shoulder just reaches 
the periphery of the blank the depth is right. The marks 
which these shoulders make on the blank, should be as nar- 
row as can be seen, when the blanks are sized right. 



81 zz. : _ y h - terete z tee.-. :o. 

Double-curve gear-cutters are designate'.! bv letters 
instead of by numbers : this is to avoid confusion in 

:: ;~e:::i_ 

F allowing is the list of epicycloidal or double-c urve 
gear-cutters : — 

^-^vf 1 ^ Cutter A :-tits 1*2 teeth. Cutter 31 :u: = 27 t: '2? teeth 

IknUe-curre .. -n .. -i •-> t; .. v" . ■">-,..- 

.. C .. U .. .. Q .. 0± .. ;- .. 

•• D ■■ 15 u • P •• 3S •• 42 • 

" E •• 16 •• • Q •■ 43 - 49 - 

u y ■• 17 " B " 50 < ; 59 " 

.. q - is •■ S " 60 "74 '■ 

« H - 19 " - T " 75 - 99 •• 

I 20 " u T • 100 ' : 149 " 

• J - 21 t: -_ V 150 " 24:9 " 

- K - 23 t: 24 ■■ TV •• 250 - Rack. 

L • 24 to 26 u X " Rack. 

A cutter that cuts more than one gear is made of 
proper form for the smallest gear in its range. Thus, 

:tttte: J ::t 21 t: 22 teeth is crrrect tor 21 teeth: 
cutter S for 60 to 74 teeth is correct for 60 teeth, 
and so on. 
E r . : ~_ :;-. :;y In ordering epicycloidal gear-cutters designate the 
Outers. letter of the cutter as in the foregoing table, also 

either give the pitch or give data that will enable us 
to determine the pitch, the same as directed for invo- 
lute cutters. 

AI ;.re care is required in making and adjusting epi- 
cycloidal gears than in making involute gears. 
^ ~-~ -" In : lering bevel gear cutters three things must s 

Ee~el (.Tear ^ c o 



./-"t:- 



1 . h er of teeth in each gear. 

2 Either the pitch of gears or the largest pitch 

dio gear; set Fig- -7 

3. The length of tooth face. 

If the shafts are not to run at right angles, it 
should be so stated, and the angle given. Involute 
cutters onlv are used for cutting bevel gears. Nc 
tempt should be made to cut epicyclodial tooth bevel gears 
with rotarv di<k cut:- 



PROVIDENCE, R. I. S5 

In ordering- worm-wheel cutters, three things must „ T How t0 order 

o ° Worm -gear 

be given : Cutters. 

1. Number of teeth in the toheel. 

2. Pitch of the worm; see Chapter XI. 

3. M^hole diameter of worm. 

In any order connected with gears or gear-cutters, 
when the word " Diameter " occurs, we usually under- 
stand that the pitch diameter is meant. "When the 
whole diameter of a gear is meant it should be plainly 
written. Care in giving an order often saves the delay 
of asking further instructions. An order for one gear- 
cutter to cut from 25 to 30 teeth cannot be filled, be- 
cause it takes two cutters of involute form to cut from 
25 to 30 teeth, and three cutters of epicycloidal form 
to cut from 25 to 30 teeth. 

Sheet zinc is convenient to sketch gears upon, and 
also for making templets. Before making sketch, it is 
well to give the zinc a dark coating with the following 
mixture : Dissolve 1 ounce of sulphate of copper (blue 
vitriol) in about 4 ounces of water, and add about one- 
half teaspoonful of nitric acid. Apply a thin coating 
with a piece of waste. 

This mixture will give a thin coating of copper to 
iron or steel, but the work should then be rubbed dry. 
Care should be taken not to leave the mixture where it 
is not wanted, as it rusts iron and steel. 

We have sometimes been asked why gears are noisy. 
Not many questions can be asked us to which we can 
give a less definite answer than to the question why 
gears are noisy. 

W T e can indicate only some of the causes that may 
make gears noisy, such as: — depth of cutting not 
right — in this particular gears are oftener cut too deep 
than not deep enough ; (more noise may be caused 
by cutting the driver too deep than by cutting the 
driven too deep;) cutting not central — this may 
make gears noisy in one direction when they are quiet 
while running in the other direction ; centre distance 
not right — if too deep the outer corners of the 
teeth in one gear may strike the fillets of the teeth 
in the other gear ; shafts not parallel ; frame of the 



B 



BE OWN i -HaF.FE MFG. CO. 

machine : : such a form as to give oft sound vibrat: : as 
;n when we examine a pair of gears e cannot 
always tell what is the matter. 

N >te — For any pitch not in the folio wing tables : 
find corresponding part : — multiply the tabular value 
for one inch by the circular pitch required, and the 
;: will be the value for the pitch given. Exam- 
ple : What is the value of s for 4 inch circular pitch? 
.3183 = $ for 1 P and .3183 ■ 4= 1.2732=8 for 4" 
P. 

The expression "Addendum and — " (addendum 

and the modules mean- the distance of a tooth outside 
of pitch line and also the distance occupied for every 
tooth upon the diameter of pitch circle. 




■■■ : = '•' ---i-z 






:i z - :- si' ~: :~ 



PART II. 



CHAPTER I. 
TANGENT OF ARC AND ANGLE. 



In Pakt II. we shall show how to calculate some e |p2fned t0 be 
of the functions of a right-angle triangle from a table 
of circular functions, the application of these calcula- 
tions in some chapters of Part I. and in sizing blanks 
and cutting teeth of spiral gears, the selection of 
cutters for spiral gears, the application of continued 
fractions to some problems in gear wheels and cutting 
odd screw-threads, etc., etc. 

A. Function is a quantity that depends upon another 
quantity for its value. Thus the amount a workman 
earns is a function of the time he has worked and of fi ^ nctl0u de ~ 
his wages per hour. 




In any right-angle triangle, O A B, we shall, for Right - angle 

t i o Triangle. 

convenience, call the two lines that form the right 
angle O A B the sides, instead of base and perpen- 
dicular. Thus O A B, being the right angle we call 
the line O A a side, and the line A B a side also. 

When we speak of the angle A O B, we call the line 
O A the side adjacent. When we are speaking of the Skle ad J acent - 
angle ABO we call the liue A B the side adjacent. 
The line opposite the right angle is the hypothenuse. Hypothenuse. 

In the following pages the definitions of circular 
functions are for angles smaller than 90°, and not 
strictly applicable to the reasoning employed in ana- 
lytical trigonometry, where we find expressions for 
angles of 270°, 760°, etc. 



88 

Tangent. 



BROWN & SHAEPE MFG. CO. 

The Tangent of an arc is the line that touches it at 
one extremity and is terminated by a line drawn from 
the center through the other extremity. The tangent 
is always outside the arc and is also perpendicular to 
the radius which meets it at the point of tangency. 




*iff ^7 



Thus, in Fig. 46, the line A B is the tangent of the arc 
A C. The point of tangency is at A. 

An angle at the center of a circle is measured by the 
arc intercepted by the sides of the angle. Hence the 
tangent A B of the arc A C is also the tangent of the 
angle A O B. 

In the tables of circular functions the radius of the 
arc is unity, or. in common practice, we take it as one 
inch. The radius O A being 1", if we know the length 
of the line or tangent A B we can, by looking in a 
table of tangents, find the number of degrees in the 
angle A O B. 
To find the Thus, if A B is 2.25" long, we find the angle A O B 

Degrees in an . ° ° 

Angle. is 66° very nearly. That is, having found that 2.2460 

is the nearest number to 2.25 in the table of tangents 
at the end of this volume, we find the corresponding 
degrees of the angle in the column at the left hand of 
the table and the minutes to be added at the top of 
the column containing the 2.2460. 

The table gives angles for every 10', which is suf- 
ficient for most purposes. 



PROVIDENCE, R. I- 89 

Now, if we have a right-angle triangle with an angle 
the same as O A B, but with O A two inches long, the 
line A B will also be twice as long as the tangent of 
angle A O B, as found in a table of tangents. 

Let us take a triangle with the side OA=5" long, fln J xa ™£% £ 
and the side AB = 8" long ; what is the number of *™ e e f in au 
degrees in the angle A O B ? 

Dividing 8" by 5 we find what would be the length 
of A B if O A was only 1" long. The quotient then 
would be the length of tangent when the radius is 1" 
long, as in the table of tangents. 8 divided by 5 is 
1.6. The nearest tangent in the table is 1.6003 and . 
the corresponding angle is 58°, which would be the 
angle A O B when A B is 8" and the radius O A is 5" 
very nearly. The difference in the angles for tangents 
1.6003 and 1.6 could hardly be seen in practice. The 
side opposite the required acute angle corresponds to 
the tangent and the side adjacent corresponds to the 
radius. Hence the rule : 

To find the tangent of either acute angle in a right- T JJ e ^J d the 
angle triangle : Divide the side opposite the angle by 
the side adjacent the angle and the quotient will be 
the tangent of the angle. This rule should be com- 
mitted to memory. Having found the tangent of the 
angle, the angle can be taken from the table of tan- 
gents. 

The complement of an angle is the remainder after complement 
subtracting the angle from 90°. Thus 40° is the com- 
plement of 50°. 

The Cotangent of an angle is the tangent of the cotangent, 
complement of the angle. Thus, in Fig. 47, the line 
A B is the cotangent of A O E. In right-angle tri- 
angles either acute angle is the complement of the 
other acute angle. Hence, if we know one acute angle, 
by subtracting this angle from 90° we get the other 
acute angle. As the arc approaches 90° the tangent 
becomes longer, and at 90° it is infinitely long. 

The sign of infinity is oo. Tangent 90° = oo, 



AMi^by 1 ^? ~^7 a ta ^ e °^ tangents, angles can be laid out npon 

Tangent Ex- 5neet zinc. ere. T_:s is often an advantage, as i: is not 
=■ - e - - ' - 

convenient to lay protractor flat down so as to mark 

angles up to a sharp point. If we could lay off the 

length of a line ~ e could take tangents dire :■: 

from table and obtain angle at moe. It. however, is 

generally bet:-: : multiply the tangent by 5 or 10 

and make an riilarged triangle. If, then, there is a 

slight error in laying off length of lines it will not 

make so much difference with the angle. 

Let it be required to lay off an angle of 14 r 30'. By 
the table we find the tangent to 1: e _ : B 1 Multiply- 
ing .25861 hj 5 we obtain, in the enlarged triangle. 
1.29305 as the length of side opposite the angle l-±~ 
30 . As we have made the side opposite five times as 
large, we must make the sid^ •int five times 

large, in order to keep angle the same. Hence. F:_ 
48, draw* the line A B 5 long;; perpendicular to this 
line at A Iraw ._e:ieA J 1.203 long; now draw the 
line O B, and the angle A B O will be 14° 30'. 

If special accuracy is required, the tangent can be 
multiplied by 10 ; the line AO will then be 2.5S6' long 
and the line A B 10 long. Remembering that the 
acute angles of a right-angle triangle are the comple- 
ments of each other, we subtract 11 3U' from 90' and 
btain 75" 30 as the angle of A O B. 

The reader will remember these angles as occurring 
in Paj.i I.. Chapter IT., and obtained in a different 
way. A Srimcirele upon the line O B touching the 
extremitic ill just touch the right angle at 

A. and the line 3:-: :ur times as long as O A 

L: it be required to turn a piece 4 long. 1 diam- 
eter at small end. with a taper of 10 r one side with the 
fcher ; what will be the diameter of the piece at th^ 
large end ? 

A section. Fig. 49, through the axis of this piece is 
To .^lcuiate l ] le saine as if we added two riorht-anofle triangles. 

Diameter of a -_ 

Tapering^ B and A' B . to a straight piecr A A B B . 1 

piece. Fig. 50. ,, =» * 

wide and 4" long,the a angles 1 an.i B bein_r 5 . 

thus making th: sides ' B and OBI with -ach 

_ 

ther. 



PROVIDENCE, R. I. 



91 




-1^293-+ 
Fig. 4S. 




Fig. 49 



92 



BROWN & SHARPE MFG. CO. 

The tangent of 5° is .08748, which, multiplied by 
4 , gives . 34992 " as the length of each line, A O and 
A' O', to be added to 1" at the large end. Taking 
twice .34992" and adding to 1" we obtain 1.69984" as 
the diameter of large end. 

This chapter must " be thoroughly studied before 
taking up the next chapters. If once the memory 
becomes confused as to the tangent and sine of an 
angle, it will take much longer to get righted than it 
will to first carefully learn to recognize the tangent 
of an angle at once. 

If one knows what the tangent is, one can tell better 
the functions that are not tangents. 




03 



CHAPTER II. 

SINE— COSINE AND SECANT : SOME OF THEIR APPLICATIONS IN 

MACHINE CONSTRUCTION. 



Sine of Arc 
and Angle. 



The Sine of an arc is the line drawn from one 
extremity of the arc to the diameter passing through 
the other extremity, the line being perpendicular to 
the diameter. 

Another definition is : The sine of an arc is the dis- 
tance of one extremity of the arc from the diameter, 
through the other extremity. 

The sine of an angle is the sine of the arc that 
measures the angle. 

In Fig. 50 , A C is the sine of the arc B C, and of 
the angle BOC. It will be seen that the sine is 
always inside of the arc, and can never be longer than 

the radius. As the arc ap- 
proaches 90°, the sine comes 
nearer to the radius, and at 90° 
the sine is equal to 1, or is the 
radius itself. From the defini- 
tion of a sine, the side A C, 
opposite the angle A O C, in 
any right-angle triangle, is the 
sine of the angle A O C, when 
O C is the radius of the arc. 
Hence the rule : In any right-angle triangle, the side to find the 
opposite either acute angle, divided by the hypothe- 
nuse, is equal to the sine of the angle. 

The quotient thus obtained is the length of side 
opposite the angle when the hypothenuse or radius is 
unity. The rule should be carefully committed to 
memory. 




Fig. 50. 



Sine. 



94 



BROWN & SHARPE MFG. CO. 



Chord of an ^ Chord is a straight line joining the extremities of 
an arc, and is twice as long as the sine of half the 
angle measured by the arc. Thus, in Fig. 51, the 
chord B C is twice as long as the sine A C. 




Fig. r>l 

Let there be four holes equidistant about a circle 
3" in diameter — Fig. 51 ; what is the shortest distance 
between two holes ? This shortest distance is the 

nnd X th™chord° chord A B, which is twice the sine of the angle COB. 
The angle A O B is one quarter of the circle, and 
C O B is one-eighth of the circle. 360°, divided by 
8=45°, the angle COB. The sine of 45° is .70710, 
which multiplied by the radius 1.5", gives length C B in tho 
circle, 3" in diameter, as 1.0G0G5". Twice this length is 
the required distance A B=2.1213". 

"When a cylindrical piece is to be cut into any num- 
ber of sides, the foregoing operation can be applied to 
obtain the w T idth of one side. A plane figure bounded 

Polygon. by straight lines is called a polygon. 



PROVIDENCE, E. I. 



95 



When the outside diameter and the number of sides of 
a regular polygon are given, to find the length of 
one of the sides: Divide 3G0° by twice the number of To fi ^, tlie 

J J length of Side. 

sides ; midtiply the sine of the quotient by the outer 
diameter, and the product will be the length of one of 
the sides. 

Multiplying by the diameter is the same as multi- 
plying by the radius, and that product again by 2. 

The Cosine of an angle is the sine of the comple- Cosine - 
ment of the angle. 

In Fig. 50, C O D is the complement of the angle 
A O C ; the line C E is the sine of COD, and hence 
is the cosine of B O C. The line O A. is equal to C E. 
It is quite as well to remember the cosine as the part 
of the radius, from the center that is cut off by the 
sine. Thus the sine A C of the angle A O C cuts off 
the cosine O A. The line O A may be called the 
cosine because it is equal to the cosine C E. 

In any right-angle triangle, the side adjacent either 
acute angle corresponds to the cosine when the 
hypothenuse is the radius of the arc that measures 
the angle ; hence: Divide the side adjacent the acute To find the 
angle by the hypothenuse, and the quotient vnll be the 
cosine of the angle. 

When a cylindrical piece is cut into a polygon of 
any number of sides, a table of cosines can be used tOg.^iverth of 
obtain the diameter across the sides. g° n - 




96 BROWN «s SHABPE MFG. CO. 

Let a cylinder, 2 " diameter. Fig. 53. be cut six-sided: 
what is the diameter across the sides ? 

The angle A O B. at the center, occupied by one of 
these sides, is one-sixth of the circle, =60 c . The 
cosine of one-half this angle, 30". is the line C O; 
twice this line is the diameter across the sides. The 
cosine of 30 : is .86602. which, multiplied by 2, gives 
1.7320-4" as the diameter across the sides. 

Of course, if the radius is other than unity, the cosine 
should be multiplied by the radius, and the product 
again by 2. in order to get diameter across the sides ; 
or what is the same thing, multiply the cosine by the 
whole diameter or the diameter across the corners, 
ameter f aoroi ^e ru - e f or obtaining the diameter across sides of 
sides of a Po1 ^- regular polvgon. when the diameter across corners i> 
given, will then be: Multiply the cosine of 360 D 
divided l>y twice the number of sides, ly the diameter 
across corners, and the product will be the diameter 
across sides. 

Look at the right-hand column for degrees of the 
cosine, and at bottom of page for minutes to add to 
the degrees. 

The Secant of an arc is a straight line drawn from 
the center through one end of an arc, and terminated 
bv a tangent drawn from the other end of the arc. 

Thus, in Fig. 53, the line OB is the secant of the 
angle. COB. 

A C B 




Fig. J3. 

To cud the In a n v light -angle triangle, divide the hypothenuse 

Secant. c 

by the side adj icent either acute angle, and the quo- 
tient will le the secant of that angle. 



PROVIDENCE, R. I. 97 

That is, if we divide the distance OB by O C, in 
the right-angle triangle COB, the quotient will be 
the secant of the angle COB. 

The secant cannot be less than the radius ; it in- 
creases as the angle increases, and at 90° the secant is 
infinity =co . 

A six-sided piece is to be H" across the sides ; towp^^^J 
large must a blank be turned before cutting the sides ? JF™ R)iygon! rS 
Dividing 360° by twice the number of sides, we have 
30°, which is the angle COB. The secant of 30° is 
1.1547. 

The radius of the six-sided piece is .75". 

Multiplying the secant 1.1547 by .75", we obtain the 
length of radius of the blank O B ; multiplying again 
by 2, we obtain the diameter 1.732" + . 

Hence, in a regular polygon, when the diameter 
across sides and the number of sides are given, to find 
diameter across corners : Multiply the secant of 360° 
divided by tvnce the number of sides, by the diameter 
across sides, and the product will be the diameter 
across corners. 

It ^Till be seen that the side taken as a divisor has 
been in each case the side corresponding to the radius 
of the arc that subtends the angle. 

The versed sine of an acute angle is the part of 
radius outside the sine, or it is the radius minus the 
cosine. Thus, in Fig. 50, the versed sine of the arc 
BC is AB. The versed sine is not given in the tables 
of circular functions : when it is wanted for any angle 
less than 90° we subtract the cosine of that angle from 
the radius 1. Having it for the radius 1, we can 
multiply by the radius of any other arc of which we 
may wish to know the versed sine. 

Fig. 54 is a sketch of a gear tooth of IP. In 
measuring gear teeth of coarse pitch it is sometimes a 
convenience to know the chordal thickness of the 
tooth, as at ATB, because it may be enough shorter 
than the regular tooth-thickness AHB, or t, to require 
attention. It may be also well to know the versed 
sine of the angle 1J, or the distance II, in order to tell 
where to measure the chordal thickness. 



98 



BliOWN & SHAKPK MI"G CO. 




PROVIDENCE, R. I. 



99 



On pages 104 and 105 are tables of data pertaining 
to chordal thickness of IP. teeth. For any other 
diametral pitch, divide the number in the tabic by that 
pitch. 




GEAR TOOTH CALIPER. 



100 



BROWN & SHAEPE MFG. CO. 



CHORDAL THICKNESS OF TEETH FOR GEARS AND CUTTERS, 
ON A BASTS OF 1 DIAMETRAL PITCH. 



T = D' sin. ff 
H = R (1— cos. £'). 



N = Number of teeth in gears. 

T = Chorclal thickness of Tooth. 

H = Height of Arc. 

D' = Pitch Diameter. 

R = Pitch Radius. 

yS f = 90° divided by the number of teeth. 

Note. — When the tooth of a gear is measured, add the height of arc to (S); and 
when gear cutter is measured subtract the height of arc from (S + f). 

Involute. 



Cutter. 


T 


H 

.0047 


Corrected 
S+f forCutt. 


Corrected 
S for Gear. 


No.l — 135 T- 


-1P 


1.5707 


1.1524 


1.0047 


« 2 — 55 T- 


-IP 


1.5706 


.0112 


1.1459 


1.0112 


u g__ 35T- 


-IP 


1.5702 


.0176 


1.1395 


1.0176 


« 4— 26T- 


-IP 


1.5698 


.0237 


1.1334 


1.0237 


«5— 21 T- 


-IP 


1.5694 


.0294 


1.1277 


1.0294 


« 6— 17 T- 


-IP 


1.5686 


.0362 


1.1209 


1.0362 


"'7— 14 T- 


-IP 


1.5675 


.0440 


1.1131 


1.0440 


«8— 12 T- 


-IP 


1.5663 


.0514 


1.1057 


1.0514 


11 T- 


-IP 


1.5654 


.0559 


1.1011 


1.0559 


10T- 


-IP 


1.5643 


.0616 


1.0955 


1.0616 


9T- 


-IP 


1.5628 , .0684 


1.0887 


1.0684 


8T- 


-IP 


1.5607 


.0769 


1.0802 


1.0769 



providence, r. i. 
Epicycloidal. 



101 



Cutter. 


T 


H 


Corrected 
S + f f or Cutt. 


Corrected 
S for Gear. 


A 


— 12T- 


-1P 


1.5663 


.0514 


1.1057 


1.0514 


B 


— 13 T- 


-IP 


1.5670 


.0474 


1.1097 


1.0474 


C - 


— 14 T- 


-IP 


1.5675 


.0440 


1.1131 


1.0440 


D 


— 15T- 


-IP 


1.5679 


.0411 


1.1160 


1.0411 


E 


— 16 T- 


-IP 


1.5683 


.0385 


1.1186 


1.0385 


F 


— 17 T- 


-IP 


1.5686 


.0362 


1.1209 


1.0362 


G 


— 18 T- 


-IP 


1.5688 


.0342 


1.1229 


1.0342 


H- 


— 19T- 


-IP 


1.5690 


.0324 


1.1247 


1.0324 


I 


— 20 T- 


-IP 


1.5692 


.0308 


1.1263 


1.0308 


J - 


— 21 T- 


-IP 


1.5694 


.0294 


1.1277 


1.0294 


K 


— 23T- 


-IP 


1.5696 


.0268 


1.1303 


1.0268 


L 


— 25 T- 


-IP 


1.5698 


.0247 


1.1324 


1.0247 


M 


— 27T- 


-IP 


1.5699 


.0228 


1.1343 


1.0228 


N- 


— 30 T- 


-IP 


1.5701 


.0208 


1.1363 


1.0208 





— 34 T- 


-IP 


1.5703 


.0181 


1.1390 


1.0181 


P 


- 38 T- 


-IP 


1.5703 


.0162 


1.1409 


1.0162 


Q 


— 43 T- 


-IP 


1.5705 


.0143 


1.1428 


1.0143 


R - 


— 50 T- 


-IP 


1.5705 


.0123 


1.1448 


1.0123 


S - 


— 60T- 


-IP 


1.5706 


.0102 


1.1469 


1.0102 


T 


- 75T- 


-IP 


1.5707 


.0083 


1.1488 


1.0083 


U- 


— 100T- 


-IP 


1.5707 


.0060 


1.1511 


1.0060 


V- 


— 150T- 


-IP 


1.5707 


.0045 


1.1526 


1.0045 


w 


— 250T- 


-IP 


1.5708 


.0025 


1.1546 


1.0025 



Special. 



No. Teeth. 



9T — IP 
10 T — IP 
11T — IP 



1.5628 
1.5643 
1.5654 



H 



.0684 
.0616 
.0559 



Corrected 
S + f for Cutt. 



1.0887 
1.0955 
1.1012 



Corrected 
S for Gear. 



1.0684 
1.0616 
1.0559 



103 



CHAPTER III. 

APPLICATION OF CIRCULAR FUNCTIONS— WHOLE DIAMETER OF 
BEVEL GEAR BLANKS— ANGLES OF BEVEL GEAR BLANKS. 



The rules given in this chapter apply onl} T to bevel 
gears having the center angle c O i not greater than 90°. 

To avoid confusion we will illustrate one gear only. 
The same rules apply to all sizes of bevel gears. Fig. 
55 is the outline of a pinion 4 P, 20 teeth, to mesh with 
a gear 28 teeth, shafts at right angles. For making 
sketch, of bevel gears see Chapter IX.. Part I. 

In Fig. 55, the line O m' m is continued to the line 
a b. The angle c O I that the cone pitch-line makes 
with the center line may be called the center angle. 
Angle of The center angle c O i is equal to the angle of edge 
' c i c. c i is the side opposite the center angle c' O 
i, and c' O is the side adjacent the center angle, c 
i = 2.5"; & O : = 3.5". Dividing 2.5" by 3.5" we 
obtain .71128" + as the tangent of c' O i. In the table 
we find .71329 to be the nearest tangent, the corre- 
sponding angle being 35° 30'. 35 |°, then, is the center 
angle c O i and the angle of edge c i n, very nearly. 

When the axes of bevel gears are at right angles the 
angle of edge of one gear is the complement of angle 
of edge of the other gear-. Subtracting, then, 35^° 
from 90° we obtain 54^° as the angle of edge of gear 
28 teeth, to mesh with gear 20 teeth, Fig. 55. from which we 
have the rule for obtaining centre angles when the axes of 
gears are at right angles. 

Divide the radius of the pinion by the radius of the gear 
and the quotient will be the tangent of centre angle of the 
pinion. 

Xow subtract this centre ande from 90 deer, and we have 
the centre angle of the gear. 

The same result is obtained by dividing the number of 
teeth in the pinion by the number of teeth in the gear ; the 
quotient is the tangent of the centre angle. 



PROVIDENCE, R. I. 



103 




Fig. 55. 

BEVEL GEAR DIAGRAM, 



104 BROWN & SHARPE MFG. CO. 

Angle of Pace. To obtain angle of face O m" c\ the distance c O 
becomes the side opposite and the distance m" c is 
the side adjacent. 

The distance c O is 3.5", the radius of the 28 tooth 
bevel gear. The distance c m" is by measurement 
2.82". 

Dividing 3.5 by 2.82 we obtain 1.2411 for tangent 
of angle of face O m" c. The nearest tangent in the 
table is 1.2422 and the corresponding angle is 51° 10'. 
To obtain cutting angle c O n" we divide the distance 
c n" by c O. By measurement c' n" is 2.2". Divid- 
ing 2.2 by 3.5 we obtain .62857 for tangent of cutting 
angle. The nearest corresponding angle in the table 
is 32°10'. 

The largest pitch diameter, kj, of a bevel gear, as in 
Fig. 56, is known the same as the pitch diameter of 
any spur gear. Now, if we know the distance b o or 
its equal a q, we can obtain the whole diameter of 
bevel gear blank by adding twice the distance b o to 
the largest pitch diameter. 
cr?ment ter Fig" Twice the distance b o, or what is the same thing, 
5b - the sum of a q and b o is called the diameter incre- 

ment, because it is the amount by Avhich we increase 
the largest pitch diameter to obtain the whole or out- 
side diameter of bevel gear blanks. The distance b o 
can be calculated without measuring the diagram. 

The angle b o j is equal to the angle of edge. 

The angle of edge, it will be remembered, is the 
angle formed by outer edge of blank or ends of teeth 
with the end of hub or a plane perpendicular to the 
axis of gear. 

The distance b o is equal to the cosine of angle of 
edge, multiplied by the distance j o. The distance j o 
is the addendum, as in previous chapters ( = s). 

Hence the rule for obtaining the diameter increment 
of any bevel gear: Multiply the cosine of angle of 
edge by the working depth of teeth (D"), and the 
product will be the diameter increment. 

By the method given on page 102 we find the angle 
of edge of gear (Fig. 56) is 56° 20'. The cosine 
of 56° 20° is .55436, which, multiplied by §", or the 
e? e U r Side Diam " depth of the 3 P gear, gives the diameter increment of 
the bevel gear 18 teeth, 3 P meshing with pinion of 12 



PROVIDENCE, R. I. 



105 




10G BROWN 6 SHABPE MFG. CO. 

teeth, j of .55436 = . 369."— I or .37". nearly). Adding 
the diameter increment. .37 . to the largest pitch 
diameter of gear, 6". we have 6.37" as the outside 
diameter. 

In the same manner, the distance c J is half the 
diameter increment of the pinion. The angle c J J: is 
equal to the center angle of pinion, and when axes are 
at right angles is the complement of center angle of 
gear. The center angle of pinion is 33° 40'. The 
cosine, multiplied by the working depth, gives .555" 
for diameter increment of pinion, and we have 4-555" 
for outside diameter of pinion. 

In turning bevel gear blanks, it is sufficiently accu- 
rate to make the diameter to the nearest hundredth of 
an inch. 
Angle in-re The small angle o / is called the angle increment. 

merit. ° f u 

When shafts are at right angles the face angle of one 
gear i 3 equal to the center angle of the other gear, 
minus the angle increment. 

Thus the angle of face of gear 'Fig. 56) is less than 
the center angle D k\ or its equal 0./ k by the angle 
o Oj. That is. subtracting o Oj from Oj A\ the re- 
mainder will be the angle of face of gear. 

Subtracting the angle increment from the center 
angle of gear, the remainder will be the cutting 
angle. 

The angle increment can be obtained by dividing 
oj. the side opposite, by Oj. the side adjacent, thus 
finding the tangent as usual. 

The length of cone-pitch line from the common 
center. t) j. can be found, without measuring dia- 
gram, by multiplying the secant of angle 0,/ A - , or the 
center angle of }:>inion, by the radius of largest pitch 
diameter of gear. 

The secant of angle Oj k, 33"" 40', is 1.2015, which. 
multiplied by 3". the radius of gear, gives 3.6045" as 
the length of line ,/. 

Dividing oj by Oj, we have for tangent .092-4. and 
for angle increment 5" 20'. 

The angle increment can also be obtained by the 
following rule : 



PROVIDENCE, E. I. 107 

Divide the sine of center angle by half the nun* 
ber of teeth, and the quotient will be the tangent of 
increment angle. 

Subtracting the angle increment from, center angles 
of gear and pinion, we have respectively : 
Cutting angle of gear, 51°. 
Cutting angle of pinion, 28° 20'. 
.Remembering that when the shafts are at right 
angles, the face angle of a gear is equal to the cutting 
angle of its mate (Chapter X. part 1), we have: 
Face angle of gear, 28° 20'. 
Face angle of pinion, 51°. 
It will be seen that both the whole diameter and the 
angles of bevel gears can be obtained without making 
a diagram. Mr. George B. Grant has made a table of 
different pairs of gears from 1 to 1 up to 10 to 1, con- 
taining diameter increments, angle increments and 
centre angles, which is published in his " Treatise on 
Gears." " Formulas in Gearing," published by us, also 
contains extensive tables for bevel gearing. We have 
adopted the terms "diameter increment," ''angle incre- 
ment," and "centre angle" from him. lie uses the 
term (i back angle" for what we have called angle of 
edge, only he measures the angle from the axis of the 

gear, instead of from the side of the gear, or from the t Tol , a y° u *?: n 
° ' & ' Angle oy the 

end of hub, as We have done ; that is, his "back angle "Sine. 
is the complement of our angle of edge. 

In laying out angles, the following method may be 




-t'i'j. 52. 



108 BROWN & SHARPE MFG. CO. 

preferred, as it does awny with the necessity of making 
aright angle: Draw a circle, ABO (Pig. o't), ten 
inches in diameter. Set the dividers to ten times the 
sine of the required angle, and point off this distance 
in the circumference as at A B. From any point O in 
the circumference, draw the lines O A and O B. The 
angle AOB13 the angle required. Thus, let the re- 
quired angle be 12°. The sine of 12° is .20791, which, 
multiplied by 10, gives 2.0791", or Zj$q-" nearly, for 
the distance A B. 

Any diameter of circle can be taken if we multiply 
the sine by the diameter, but 10" is very convenient, 
as- all we have to do with the sine is to move the 
decimal point one place to the right. 

If either of the lines pass through the centre, then the 
two lines which clo not pass through the centre will form a 
right angle. Thus, if O B passes through the centre then 
the two lines A B and A O will form a right angle at A. 



109 



CHAPTER IV. 
SPIRAL GEARS — CALCULATIOHS FOR PITCH OF SPIRALS. 



When the teeth of a gear are cut, not in a straight s P iral Gear « 
path, like a spur gear, but in a helical or screw- like 
path, the gear is called, t actinically, a twisted or screw 
gear, but more generally among mechanics, a spiral 
gear. A distinction is sometimes made between a 
screw gear and a twisted gear. In twisted gears the 
pitch surfaces roll upon each other, exactly like spur 
gears, the axes being parallel, the same as in Fig. 1, 
Part I. In screw gears there is an end movement, 
or slipping of the pitch surfaces upon each other, the 
axes not being parallel. In screw gearing the action 
is analogous to a screw and nut, one gear driving 
another by the end movement of its tooth path. This 
is readily seen in the case of a worm and worm-wheel, 
when the axes are at right angles, as the movement of 
wheel is then wholly due to the end movement of 
worm thread. But, as we make the axes of gears more 
nearly parallel, they may still be screw gears, but the 
distinction is not so readily seen. 

We can have two gears that are alike run together, 
with their axes at right angles, as at A B, Fig. 59. 

The same gear may be used in a train of screw gears 
or in a train of twisted gears. Thus, B, as it relates to 
A, may be called a screw gear; but in connection with 
C, the same gear, B, may be called a twisted gear. 
These distinctions are not usually made, and we call 
all helical or screw-like gears made on the Universal 
Milling Machine spiral gears. 

When two extarnal spiral gears run together, with Direction of. 
their axes parallel, the teeth of the gears must have erence to Axes. 
opposite hand spirals. 



110 



BROWN & SHAKPE MFG. CO. 



Thus, in Fig. 59 the gear B has right hand spiral 
teeth, and the gear C has left hand spiral teeth. When 
the axes of two spiral gears are at right angles, both 
gears must have the same hand spiral teeth. A and 
B, Fig. 59. have right hand spiral teeth. If both gears 
A and B had left hand spiral teeth, the relative direc- 
tion in which they turn would be reversed, 
spiral Lead. ^he spiral lead or lead of spiral is the distance the 
spiial advances in one turn. A cylinder or gear cut 
with spiral grooves is morel)' a screw of coarse pitch or 
long lead ; that is, a spiral is a coarse lead screw, and 
a screw is a fine lead spiral. 

Since the introduction and extensive use of the 
Universal Milling Machine, it lias become customary 
to call any screw cut in the milling machine a spiral. 
The spiral lead is given as so many inches to one 
turn. Thus, a cylinder having a spiral groove that 
advances six inches to one turn, is said to have a six 
inch spiral. 

In screws the pitch is often given as so many turns 
to one inch. Thus, a screw of -J-" lead is said to be 2 
turns to the inch. The reciprocal expression is not 
much used with spirals. For example, it would not 
be convenient to speak of a spiral of 6" lead, as -J- turns 
to one inch. 

The calculations for spirals are made from the func- 
tions of a right angle triangle. 
Example, Cut from paper a right angle triangle, one side of 

showing the r l ° ° . 

nature of a He- the right angle 6" long, and the other side of the 

lix or Spiral. c " c 

right angle 2 . Make a cylinder 6 in circumference. 
It will be remembered (Part I., Chapter II.) that the 
circumference of a cylinder, multiplied by .3183, equals 
the diameter — 6" X .3183=1.9098". Wrap the paper 
triangle around the cylinder, letting the 2" side be 
parallel to the axis, the 6" side perj)endicular to the 
axis and reaching around the cylinder. The hypoth- 
eneuse now forms a helix or screw-like line, called 
a spiral. Fasten the paper triangle thus wrapped 
around. See Fig. GO. 



PROVIDENCE, R. I. 



Ill 




FIG. 58 -RACKS AND GEARS. 




Fig. 59.-SPIRAL GEARING. 



11*2 



i ?. ; ~>~ 



: z vy- :: 




Wig SO 



If we dow turn this cylinder ABCD one lorn in 
the direction of the arrow, the spiral will adva nee from 
to E. This advai.ee is the lead of the sp 

The angle E F, which the spiral makes with the 
axis E 3 is the angle of the spiral. This ;mgle isfonnd 
as in Chapter I. The circumference of the cylinder 

:;s: :n::s ::■ t :- side opposite the angle. T . -:■ pilch 
of the spiral corresponds to the si seat the angle. 

Hence the role for angle of spiral: 
elating tie Divide tlte circumference of the 
parts of a spi -fry f] te numo er of inches of 5 the 

quotient will he the tangent of angle - 

When the angk >fspi and : '.inference are given, 
to find the lead : 

DivU/e the circrr/i/'trence by the tangent of angle, 
the quotient will be the lead of the spi 

When the angle of spiral and the lead or pitch of spiral 
are sriven, to find, the ci : rence : 

MuVij the t gent of angle by the lead, and the 
product will be t ce. 

When applying calculations to spiral gears the angle 
is reckoned at the pitch circumference and not at the 
outer or addendum circle. 

It will be seen that when two spirals of different 
diameters lave the b the spiral of less diame- 

ter will have the smaller angle. Thus in Fig. GO if ihe 
paper triangle bid teen 4" lonsr ii s: of C" the diam- 
eter of t e cylinder would have been 1.27" .d the 
angle of the spiral would have been _ legrees. 



113 



CHAPTER V. 

EXAMPLES IN CALCULATION OF THE LEAD OF SPIRAL— ANGLE OF 

SPIRAL— CIRCUMFERENCE OF SPIRAL GEARS— 

A FEW HINTS ON CUTTING. 



It will be seen that the rules for calculating the cir- 
cumference of spiral gears, angle and the lead of spiral 
are the same as in Chapter I., for the tangent and angle 
of a right angle triangle. In Chapter IV., the word 
"circumference" is substituted for "side opposite," 
and the words "lead of spiral" are substituted for 
"side adjacent." 

When two spiral gears are in mesh the angle of r ^ 1 ff^ ( ^ e f JjJ" 
spiral should be the same in one gear as in the other, *f C Q h l f t An s le 
in order to have the shafts parallel and the teeth work 
properly together. When two gears both have right 
hand spiral teeth, or both have left hand spiral teeth, 
the angle of their shafts will be equal to the sum of 
the angles of their spirals. But when two gears have 
different hand spirals the angle of then* shafts will be 
equal to the difference of their angles of spirals. 
Thus, in Fig. 59 the gears A and B both have right 
hand spirals. The angle of both spirals is 45°, their 
sum is 90°, or their axes are at right angles. But C 
has a left hand spiral of 45°. Hence, as the difference 
between angles of spirals of B and C is 0, their axes 
are parallel. 

If two 45° gears of the same diameter have the same 
number of teeth the lead of the spiral will be alike in 
both gears: if one gear has more teeth than the other 
the lead of spiral in the larger gear should be longer 
in the same ratio. Thus, if one of these gears has 50 
teeth and the other has 25 teeth, the lead of spiral ra ^ s ea d f ^lil^ 
in the 50 tooth gear should be twice as long as that of ent diameters. 
the 25 tooth gear. Of course, the diameter of pitch 



114: KBOWH £ SHARPE MFG. CD. 

circle should s a large in the 50 tooth as in the 

"25 tooth gear. 

In spirals where the angle is 45" the circumference 
is the same as the spiral lead, because the tangent of 
45° is 1. 
oiS2S? on *** Sometimes the circumference is varied to suit a pitch 

v, 1 rPTl TO TPTPT1 Of* -I 

- "-" ■- Si nraL that can be cut on the machine and retain the angle 
required. This would apply to catting rolls for mak- 
ing diamond-shaped impressions where the diameter 
of the roll is not a matter of importance. 

When two gears are to run together in a given 
velocity ratio, it is well first to select spirals that the 
machine will cut of the same ratio, and calculate the 
numbers of teeth and angle to correspond. This will 
often save considerable time in fisfurinof. 

The calculations for spiral gears present no special 
difficulties, but sometimes a little ingenuity is required 
to make work conform n the machine and to such 
cutters as we may have in stock. 

Let it be required to make two spiral gears to run 
with a ratio of 4 to 1, the distance between centres 
be 3.125" m"), the axes to be parallel. 

By rule given in Chapter XII.. Part I., we fiud the 
diameters of tch : rclea will be 5" and 1\". Let us 
take a spiral of 4S" lead for the large gear, and a 
spiral of 12" lead for the small gear. The circumfer- 
ence of the 5" pitch circle is 15.70796". Dividing 
the circumference by the lend of the spiral, we have 
1 5 4g 7 9 6 =-32724" for tangent of angle of spiral. In 
the ta 3 the :.■:. rest angle to tangent, .32724", is 1S° 1 1 ' . 

As before stated, the angle of the teeth in the small 
gear will be the same as the angle of teeth or spiral in 
the large gear. 
inAiigtesafctop Now, this rule gives the angle at the pitch surface 
spSjlGr^oave? on ^• I^P 0U looking at a small screw of coarse pitch, 
it will be seen that the angle at bottom of the thread 
is not so great as the angle at top of thread; that is, 
the thread at bottom is nearer parallel to the centre 
line than that at the top. 

This will be seen in Fig. 61, where A is the centre 
line; i f shows lirectiou of bottom of thread, and d g 



PROVIDENCE, K. I. 



115 



shows direction of top of thread. The angle Afb is 

less than the angle A g d. The difference of angle 

being dne to the warped nature of a screw thread. 

A cylinder 2" diameter is to have spiral grooves 20° Example in 
J ib calculation of 

with the centre line of cylinder; what will be the lead Leadof spiral. 
of spiral? The circumference is 6.2832". The tan- 
gent of 20° is .36397. Dividing the circumference by 
the tangent of angle, we obtain ^: § f § § 7 — 17.26"+ for 
lead of spiral. 




Fig. 01. 



116 



CHAPTER VI. 

NORMAL PITCH OF SPIRAL GEARS— CURVATURE OF PITCH 
SURFACE— FORM OF CUTTERS. 



Normal to a ^ Normal to a curve is a line perpendicular to the 
tangent at the point of tangency. 




In Fig. 62, the line B C is tangent to the arc DEF, 
and the line A E O, being perpendicular to the tan- 
gent at E the point of tangency, is a normal to the 
arc. 

Fig. 63 is a representation of the pitch surface of a 
spiral gear. A' D' C is the circular pitch, as in Part 
I. A D C is the same circular pitch seen upon the 
periphery of a wheel. Let A D be a tooth D and a 
space. Now, to cut this space D C, the path of cut- 
ting is along the dotted line a b. By mere inspection, 
we can see that the shortest distance between two 
teeth along the pitch surface is not the distance 
ADC. 

Let the line A E B be perpendicular to the sides of 
teeth. upon the pitch surface. A continuation of this 
line, perpendicular to all the teeth, is called the 
Normal Helix. The line A E B, reaching over a 
tooth and a space along the normal helix, is called the 
Normal Pitch. 



PUOVIDENCK, K. I. 



n 




Fig. 63. 



IIS " ~-'- -~>" : -:~z :■:?:- : 

Roma: pitch. Xhe Xormal Fitch of a spiral gear is then: _ 
shortest distance betwee the centers of ' 

:-:-;-\ ■ - :r..<u •-.■ 1 ■:>.'.: ..' : : <- p'::\ >:. ;~h-\ 

In spur gears the normal pitch and circular pitch 
are alike. In the rack D D. Fig ' 8, the linear pitch 
and normal pitch are alike. 
ltter for From the foregoing it will be seen that, if we should 
at the Bp:v:e D C with :-. ::er 3 the thickness of which 
at the pitch line is equal to one-half the circular pitch. 
a? in spur ^h—is. the space would be too - ide, and 
the teeth would be too thin. Hence, spiral gears 
should be it with thinner cutters than spur gears of 
the same circular pitch. 

The aztgie CAB is equal to the angle of the spiral. 

The line AEBc rrresponds to the cosine of the angle 

C A B. Hence the rule : Mu It y lythe - gle 

mlTpPeL^ * °f *P* ra ? ty the t far pitch, t d he product will be 

at pitch. One-half the normal pitch is the 
proper thickness of cutter at the pitch line. 

If the normal pitch and the angle are kno~ L vide 

e normal pitch by the cost <f he angle and the quo- 
: be Hie lint itch. 

This may be required iu a case of a spiral ] ion i on- 
ning in a rack. The perpendicular to the side of the 
rack is taken as the line from which to calculate angle 
of teeth. That is, this line would correspond to the 
axial line in a spiral gear; and, when the a: ::s of the 
geai is at right angles t t".:e rack, the angle of 
:e ;■:_ ~ .".h :'..e s:de of the ri.ck is -: rai r ~. ysubtruit- 
iug this angle from 90°. 

The angle of the r : ; teeth with the side of the 
rack can also be : : ained by remembering that the 
: sin :- of the angle of spiral is the sine of :_e angle of 
he teeth with the s i :1 e of the rack. 

The addendum and working dep:h of tooth sh: 
correspond to the >al pitch, and not to the circular 

tch. Thus, if the normal pitch is 12 di the 

addendum should be -jV', the t hkness .13 hi'". 
on. The di -. of pitch circle of a spiral gear is 
calculated fr m the etral pitch. Thus a gear of 

30 teeth 10 P w raid .itch diametr . 



Normal Pitch 
varies. 



PROVIDENCE, R. I. 119 

But il the normal pitch is 12 diametral pitch, the 
blank will be 3 T y diameter instead of 3 r y. 

It is evident that the normal pitch varies with the 
angle of spiral. The cutter should be for the normal 
pitch. In designing spiral gears, it is well first to look 
over list of cutters on hand, and see whether there are 
cutters to which the gears can be made to conform. 
This may avoid the necessity of getting a new cutter, 
or of changing both drawing and gears after they are 
under way. To do this, the problem is worked the 
reverse of the foregoing; that is: 

First calculate to the next finer pitch cutter than gl e ?sp e irai 
would be required for the diametral pitch. cutter^^lven 

Let us take, for example, two gears 10 pitch and 30 
teeth, spiral and axes parallel. Let the next finer cut- 
ter be for 12 pitch gears. The first thing is to find the 
angle that will make the normal pitch .2G18", when the 
circular pitch is .3142". See table of tooth parts. 
This means (Fig. 63) that the line A D C will be .3142" 
when A E B is .2018". Dividing .2015" by .3142" (see 
Olnip. IV.), we obtain the cosine of the angle CAB, 
which is also the angle of the spiral, ;§f J§=.833. 

The same quotient comes by dividing 10 by 12, 
yf =.833 + ; that is, divide one pitch by the other, the 
larger number being the divisor. Looking in the table, 
we find the angle corresponding to the cosine .833 is 
33° 30'. We now want to find the pitch of spiral that 
will give angle of 33 1-° on the pitch sui face of the wheel, 
3" diameter. Dividing the circumference by the tan- 
gent of angle, we obtain the pitch of spiral (see Chap. 
V.) The circumference is 9.4248". The tangent of 
33° 30' is .06188, £;f 1^= 14.23 ; and we have for 
our spiral 14.23" lead. 

When the machine is not arranged for the exact When exact 

• p •! ' ii Pitch cannot b© 

pitch of spn*al wanted, it is generally well enough to cut. 
take the next nearest spiral. A half of an inch more 
or less in a spiral 10" pitch or more would hardly be 
noticed in angle of teeth. It is generally better to 
take the next longer spiral and cut enough deeper to 
bring center distances right. When two gears of the 
same size are in mesh with their axes parallel, a change 



120 



BROWS .s SHAKPE MFG. CO. 



of angle of teeth or spiral makes no difference in the 
correct meshing < f the teeth. 
&&** Gears g nt when gears of different size are in mesh, clue 

of Different & 

sizes of Mesh. re g ar< j must be had to the spirals being in pitch, pro- 
portional to then angular velocities (see Chapter V. ) 
We come now to the curvature of cutters for spiral 
gears; that is, then shape as to whether a cutter is 
made to cut 12 teeth or 100 teeth. A cutter that is light, 
Shape of cut- to cut a spur gear 3 diameter, may not be right for a 
spiral gear 3" diameter. To find the curvature of 
cutter, fit a templet to the blank along the line of the 
normal helix, as A E B. letting the templet reach over 
about two or three normal pitches. The curvature of 
this templet will be nearer a straight line than an arc 
of the addendum circle. Xow find the diameter of a 
circle that will approximately fit this templet, and con- 
eider this circle as the addendum circle of a gear for 
which we are to s lect a cutter, reckoning the gear as 
of a pitch the fame as the normal pitch. 




Fig 64 



Thus, in Fig. 64. suppose the templet fits a circle 
3^-" diameter, if the normal pitch is 12 to inch, dia- 
metral, the cutter required is for 12 P and 40 teeth. 
The curvature of the templet will not be quite circular, 
but is sufficiently near for practical purposes. Strictly, 



• PROVIDENCE, R. I. 121 

a, flat templet cannot be made to coincide with the 
normal helix for any distance whatever, but any greater 
refinement than we have suggested can hardly be car- 
ried out in a workshop. 

This applies more to an end cutter, for a disk cutter 
may have the right shape for a tooth space and still 
round off the teeth too much on account of the warped 
nature of the teeth. 

The difference between normal pitch and linear or 
circular pitch is plainly seen in Figs 58 and 59. 

The rack T> D, Fig. 58, is of regular form, the depth 
of teeth being -^ of the circular pitch, nearly (.6866 of 
the pitch, accurately). If a section of a tooth in either 
of the gears be made square across the tooth, that is a 
normal section , the depth of the tooth will have the 
same relation to the thickness of the tooth as in the 
rack just named. 

But the teeth of spiral gears, looking at them upon 
the side of the gears, are thicker in proportion to their 
depth, as in Fig. 59 This difference is seen between 
the teeth of the two racks D D and E E, Fig. 58. In 
the rack D D we have 20 teeth, w T hile in the rack E E 
we have but 14 teeth ; yet each rack will run with each 
of the spiral gears A, B or C, Fig. 59, but at different 
angles. 

The teeth of one rack will accurately fit the teeth of 
the other rack face to face, but the sides of one rack 
will then be at an angle of 45° with the sides of the 
other rack. At F is a guide for holding a rack m mesh 
with a gear. 

The reason the racks will each run with either of the 
three gears is because all the gears and racks have the 
same normal pitch. When the spiral gears are to run 
together they must both have the same normal pitch. 
Hence, two spiral gears may run correctly together 
though the circular pitch of one gear is not like the 
circular pitch of the other gear. 

If a rack is to run at any angle other than 90° with 
the axis of the gear it is well to determine the data 
from a diagram, as it is very difficult to figure the 
angles and sizes of the teeth without a sketch or 
diagram. 



122 



CHAPTER VII. 
CUTTING SPIRAL GEARS IN A UNIVERSAL MILLING MACHINE. 



A rotary disk cutter is generally preferable to a shank 
cotter or end mill on account of cutting faster and hold- 
ing its shape longer. In catting spiral grooves, it is 
sometimes necessary to use an end mill on account of 
the warped character of the grooves, but it is very Sel- 
dom necessary to use an end mill in cutting spiral gears. 
Proving the Before catting into a blank it is well to make a slight 

Setting "f the ° c 

Machine. trace of the spiral with the cutter, after the change 
sears are in place, to see whether the gears are correct. 
If the material of the gear blanks is quite expensive, it 
is a safe plan to make trial blanks of cast iron in order 
to prove the setting of the machine, before cutting into 
the expensive material. 

The cutting of spiral gears may develop some curi- 
ous facts to one that has not studied warped surfaces. 
The gears. Fig. 59, were cut with a planing tool in a 
shaper, the spiral gear mechanism of a Universal Mill- 
ing Machine having been faste. el upon the shaper. 
The tool was of the same form as the spaces in tl 
D D, Fig. 58. Ail spiral gears of the same pitch can be 
cut in this manner with one tool. The nature of tl is 
cutting operation can be understood from a considera- 
tion of the meshing of straight side rack teeth with a 
spiral gear, as in Fig. 58. Spiral gears that run cor- 
rectly with a rack, as in Fig. 5S, will run correctly 
with each other when tin. ir axes are parallel, as at B C, 
Fig. 59 j but it is not considered that they are quite 
correct, theoretically, to run together when the gears 
have the same hand spiral, and their axes are at right 



PKOVIDENCE, II. I. 



12 



> Q 



B 



Fig. 65 



C 



k 


'rrn- 


/ 


tti 


? 




\ 








« 






\ 






/ 


\ 



_Fi<7. ec 



:u 



31. ~> - szi_?.ji ::e 



angles, as A B . Fig. ' \ thoush tbev will ran we 

:::::■. .It. Tie c^.^::.. c: c:::::g s : : i- 

a planer tool is sometimes caHlodplanimg the tee: 
log is an accurate way of shaping teeth th 
with rack teeth and for gears on parallel si 
Lr:'.::lh:.si.7r , j ez; i : -■.:". :: c •:.: s"i: 1 ti 
c.'.re p'ner i./o-6=, :: . : his r.:: .eer. :::m: 
::; geie :..". use. 

I: is c:Lve::t!i: :: live tie 5i:i :_ si 
. t .. : s :i :ne ::^:~.ii i..b.e : 



h. p:. 



_ __ j. _ __, 





-ion. 


No. of Teeth 




Pitch Diam 


Eft 1 


. 




<~ iiside Lis 


i_e:e: 


. 




Circular Pit 


:ch 






Alible of Teeth with Axis 






N nnal Cin 


c-nlar Pitch 






P:::L :■: Ci 


truer . 








A lie- ivr ■ 


s 




. 




Tni?k-ess . 


: Tooth t 








Wh-> I'er 


:LI» -: 








>\ : 


er 








- act Lead 


of Spiral 








A] _ : rin 


:e Lei i : : Si :: - 






1 - - \ rs n ~\T : 


lhng Machine to Cut Spiral 




Gear on W< 






1st Gear on 


>r: :1 




2nd Gear 01 


*v - - - 




't-s.: :i_ .^;r 


e" 7 ^" 





A spiral of any angle I : - : : can generally be cat in 
a Universal Milling Machine without special attach- 
ments, the cntter being at the top of the work. The 
cutter is placed on the arbor in such position tha: 
can reach the work centrally after the table is s : it :o 
the angle of the spiral. In order to cut central, it ia 
generally well enough to place the table, before setting 
it to the angle, so that the work centres will be central 
with the cutter, then swing the table and set it to the 
ui: :- : t'.e s ; ir;-/ . 

For very accurate work, i: is s\fer to test the posi-^^ 1 *** 1 Set * 
tion of the centres after the table has been set to the 
angle. 



PROVIDENCE, R. I. 



125 




Fig. 67. 



USE OF VERTICAL SPINDLE MILLING ATTACHMENT IN CUTTING 

SPIRAL GEARS. 



120 BROWN & SHARPE MFG. CO. 

This can be done with a trial piece, Fig. 65 ? which 
is simply a round arbor with centre holes in the ends. 
It is mounted between the centres, and the knee is 
raised until the cutter sinks a small gash, as at A. 
This gash shows the position of the cutter; ami if the 
gash is central with the trial piece, the cutter will be 
central with the work. If preferred, the arbor can be 
dogged to the work spindle; and the line B C drawn 
on the side of the arbor at the same height as the cen- 
tres ; the work spindle should then be turned quarter 
way round in order to bring the line at the top. The 
gash A can now be cut and its position determined with 
the line. 

In cutting small gears the arbor can be dogged to the 
work spindle ; the distance between the gear blank and 
the dog should be enough to let the dog pass the cutter 
arbor without striking. 

A spiral gear is much more likely to slip in cutting 
than a spur gear. 

For gears more than three or four inches in diameter 
it is well to have a taper shank arbor held directlv in 
the work spindle, as shown in Figs. 67 and GS ; and for 
the heaviest work, the arbor can be drawn into the spin- 
dle with a screw in a threaded hole in the end of the 
shank. 

After cutting a space the work can be dropped away 
from the cutter, in order to avoid scratching it when 
coming back for another cut. Some workmen prefer 
not to drop the work away, but to stop the cutter and 
turn it to a position in which its teeth will not touch 
the work. To make sure of finding a place in the cut- 
ter that will not scratch, a tooth has sometimes been 
t iken out of the cutter, but this is not recommended. 
The safest plan is to drop the work away. 
Angie^reater In cutting spiral gears of greater angle than 45°, a 
vertical spindle milling attachment is available, as 
shown in Figs. G7 and 6$. 

In Fig. G7 the cutter is at 90° with the work spindle 
when the table is set to 0, so that the proper angle at 
which the table should be set, is the difference between 
the angle of the spiral and 90°. Thus, to cut a 70° 



PROVIDENCE, R. I. 



127 




Fig. 68. 



USE OF VERTICAL SPINDLE MILLING ATTACHMENT IN CUTTING 

SPIRAL GEARS. 



128 bbow>" fc ^:-:aspe mt 

spiral, we subtract ?i : from £ . : . and the remainder, 
20°, is the angle to se: the :. ... In cutting on the 
top, Fig. 67, the attachment is set to 0. 

In Fig. 69 the cutter is at the side of the work; the 
h ile is set to 0, and the attachment is set to the differ- 
e :>:: between 9 1 : an :I the r-: : .ire 1 ^n_'e ci spiral. 

In setting the cotter central it is convenient :: have a 
small knee as at K. Fig. 66. A line is drawn uponthc 
knee at Lhe same height as : the cent es. The cutter 
art 9i is . : glit to tl. : angle as just shown, and a gash 
is cut in the knee. Wh n the g ish is central with the 
lit r. the cutter wiD be central with the work. 

The cutter can be sir to act apon either side :: the 
gear to be : ". :::r ling as a right hand or a left hand 
spiral is wanted. The setting in Fig. 68 is for a right 
hand spiral. 

If the gear blank were bronght in front of the c ::- 
t:-r. and the reversing gen: s.t between two chance 
geai a, the machine w . al .1 be 5t: f« r a left hand spiral. 

For cc iraer pitches than abon: 12 IP diametral, it is 
well to cut more than once around, the finishing cut 
being quite light so as to en: sn;::th. 



129 



CHAPTER VIII. 
SCREW GEARS AND SPIRAL GEARS— GENERAL REMARKS. 



The working of spiral gears, when their axes are working of 
., , . n iii * Spiral Gears. 

para. lei, is generally smoother than spur gears. A 

tooth does not strike along its whole face or length at 

once. Tooth contact first takes place at one side of the 

gear, passes across the face and ceases at the other 

side of the gear. This action tends to cover defects in 

shape of teeth and the adjustment of centres. 

Since the invention of machines for producing accu- 
rate epicyloidal and involute curves, it has not so often 
been found necessary to resort to spiral gears for 
smoothness of action. A greater range can be had in 
the adjustment of centers in spiral gears than in spur 
gears. The angle of the teeth should be enough, so 
that one pair of teeth will not part contact at one side 
of the gears until the next pah' of teeth have met on the 
other side of the gears. When this is done the gears 
will be in mesh so long as the circumferences of their 
addendum circles intersect each other. This is some- 
times necessary in roll gears. 

Relative to spur and bevel gears in Part I., Chapter 
XII., it was stated that all gears finally wore them- 
selves out of shape and might become noisy. Spiral 
gears may be worn out of shape, but the smoothness 
of action can hardly be impaired so long as there are 
any teeth left. For every quantity of wear, of course, 
there will be an equal quantity of backlash, so that if 
gears have to be reversed the lost motion in spiral 
gears will be as much as in any gears, and may be 
more if there is end play of the shafts. In spiral gears End Pressure 
there is end pressure upon the shafts, because of the Spiral Gears, 
screw-like action of the teeth. This end pressure is 
sometimes balanced by putting two gears upon each 
shaft, one of right and one of left hand spiral. 



130 BROWN & SHARPE MFG. CO. 

The same result is obtained in solid cast gears by 
making the pattern in two parts — one right and one 
left-hand spiral. Such gears are colloquially called 
"herring-bone gears.'' 

In an internal spiral gear and its pinion, the spirals 
of both wheels are either right-handed or left-handed. 
Such a combination would hardly be a mercantile 
product, although interesting as mechanical feat. 

In screw or worm-gears the axes are generally at 
right angles, or nearly so. The distinctive features of 
screw gearing may be stated as follows : 

The relative angular velocities do not depend upon 

the diameters of pitch- cylinders, as in Chapter I., 

Distinctive Part I. Thus the worm in Chapter XL, Fig-. 35, can 

teatures of x ' ° ' 

Screw Gearing, be any diameter — one inch or ten inches — without 
affecting the velocity of the worm-wheel. Conversely if the 
axes are not parallel we can have a pair of spiral or screw 
gears of the same diameter, but of different numbers of 
teeth. The direction in which a worm-wheel turns depends 
upon whether the worm has a right-hand or left-hand thread. 
When angles of axes of worm and worm-wheel are 
oblique, there is a practical limit to the directional 
relation of the worm-wheel. The rotation of the 
worm-wheel is made by the end movement of the 
worm-thread. 

The term worm and worm-wheel, or worm -gearing, 
is applied to cases where the worms are cut in a lathe, 
and the shapes of the threads or teeth, in axial section, 
are like a rack. The shape usually selected is like the 
rack for a single curve or involute gear. See Chap. 
IV., Parti. Worms are sometimes cut in a milling 
machine. 

If the form of the teeth in a pair of screw gears is 
determined upon the normal helix, as in Chap. VI., 
the gears are usually called Spiral Gears. 

If we let two cylinders touch each other, their axes 
being at right angles, the rotation of one cylinder will 
have no tendency to turn the other cylinder, as in 
Chapter L, Part I. 



PROVIDENCE, R. I. 13J 

We can now see why worms and worm-wheels wear why Wor ™ 
J Wheels wear 

out faster than other gearing. The length of worm- sofast - 
thread, equal to more than the entire circumference of 
worm, comes in sliding contact with each tooth of the 
wheel during one turn of the wheel. 

The angle of a worm-thread can be calculated the 
same as the angle of teeth of spiral gear ; only, the 
angle of a worm thread is measured from a line or 
plane that is perpendicular to the axis of the worm. 



CHAPTER IX. 

CONTINUED FRACTIONS— SOME APPLICATIONS IN MACHINE 

CONSTRUCTION. 



Definition of _^_ continued fraction is one which has unitv for its 

a Continued 

Fraction. numerator, and for its denominator an entire number 

plus a fraction, which fraction has also unity for its 

numerator, and for its denominator an entire number 

plus a fraction, and thus in order. 

The expression. * 

4 — i_ 

3 — 1_ 

5 is called a continued frac- 
tion. By the use of continued fractions, we are ena- 
Pracucai u=e D ; e d to nnd a fraction expressed in smaller numbers. 

of Continued *- 

Fractions. that, for practical purposes, may be sufficiently near in 
value to another fraction expressed in large numbers. 
If we were required to cut a worm that would mesh 
with a gear -4 diametral pitch -i P. . m a lathe having 
3 to 1-inch linear leading screw, we might, without 
continued fractions, have trouble in finding- change 
gears, because the circular pitch corresponding to 
•4 diametral pitch is expressed in large numbers : 

This example will be considered farther on. For 
illustration, we will take a simpler example. 

What fraction expressed in smaller numbers is near- 
est in value to T : -- '■ Dividing the numerator and the 
denominator of a fraction by the same number does 
not change the value of the fraction. Dividing both 
Example in terms f s;_ bv 29. we have ^T~ or. what is the 

Continued u« .. " J T ~ ' 

same thing expressed as a continued fraction, s-kju The 

continued fraction s+j_ is exactly equal to -f£j- ^ 
now. we reject the v 7 , the fraction 4- will be larger 
than 5— :_. because the denominator has been diniin- 

2 9' 

islied, 5 beii _ less than 5gV- i ' d something 

^-_V expressed in smaller numbers than 29 for a 



PROVIDENCE, E. I. 133 

numerator and 146 for a denominator. Eeducing \ 
and y 2 ^ to a common denominator, we have ^=i|f 
an d i 2_9_ = ±|i-. Subtracting one from the other, we 
have t Jq, which is the difference between \ and -f£-$. 
Thus, in thinking of T % as £, we have a pretty fair 
idea of its value. 

There are fourteen fractions with terms smaller than 
29 and 146, which are nearer j 2 ¥ V than \ is, such as 
■if, -if and so on to f 2 f T . In this case by continued frac- 
tions we obtain only one approximation, namely -J, and 
any other approximations, as T f, -J-f, &c, we find by 
trial. It will be noted that all these approximations 
are smaller in value than T 2 I %. There are cases, how- 
ever, in which we can, by continued fractions, obtain 
approximations both greater and less than the required 
fraction, and these will be the nearest possible approxi- 
mations that there can be in smaller terms than the 
given fraction. 

In the French metric system, a millimetre is equal 
to .03937 inch ; what fraction in smaller terms ex- 
presses .03937" nearly? .03937, in a vulgar fraction, 
is ToSood* Dividing both numerator and denominator 
by 3937, we have 25II1JL, Rejecting from the de- 
nominator of the new fraction, -Jf Jf , the fraction -fa 
gives us a pretty good idea of the value of .03937". 
If in the expression, Tz+TJJA, we divide both terms of 
the fraction |jj|j by 1575, the value will not be changed. 
Performing the division, we have 1 



25 + 1 



2 + 787 
1575 • 

"We can now divide both terms of r \ 8 T 7 T by 787, 
without changing its value, and then substitute the 
new fraction for ^ 8 T \ * n ^ ne continued fraction. 

Dividing again, and substituting, we have : 
1 

25 + 1 

2 + l_ 

2+ 1 
787 

as the continued fraction that is exactly equal to 
.03937. 



134 BROWN & SHAEPE MFG. CO. 

In performing the divisions, the work stands thus 

3937) 100000 (25 
7874 

21260 
19685 



1575) 3937 (2 
3150 

787) 1575 (2 * 

1574 

1) 787 (787 
787 

•o- 
That is, dividing- the last divisor by the last remain- 
der, as in finding the greatest common divisor. The 
quotients become the denominators of the continued 
fraction, with unity for numerators. The denominators 
25, 2, and so on, are called incomplete quotients, since 
they are only the entire parts of each quotient. The 
first expression in the continued fraction is ^ or 
.04 — a little larger than .03937. If, now, we take 
25 + i , we shall come still nearer .03937. The expres- 
sion 25~+t is merely stating that 1 is to be divided by 
25^. To divide, we first reduce 25J to an improper 
fraction, ^, and the expression becomes IT, or one 

divided by ^-. To divide by a fraction, "Invert the 
divisor, and proceed as in multiplication." We 
then have -f T as the next nearest fraction to .03937. 
-g2 T - 0392 + , which is smaller than .03937. To get still 
nearer, we take in the next part of the continued frac- 
tion, and have 1 

25 + 1 



2 + 1 
2" 

We can bring the value of this expression into a 
fraction, with only one number for its numerator and 
one number for its denominator, by performing the 
operations indicated, step by step, commencing at the 
last part of the continued fraction. Thus, 2 + J, or 
2-|, is equal to ■§-, Stopping here, the continued frac- 
tion would become } 

25 -H_ 

2- 

1 1__ 

Now, 5 equals -|, and we have 25 +_»_. 25f equals 

2 5 

1 \ 3 -; substituting again, we have rhi. Dividing 1 by 
J-f- 1 , w^e have T | T . T | T is the nearest fraction to 



PROVIDENCE, E. I. 135 

.03937, unless we reduce the whole continued fraction 
l 

25+1 

2 + 1 

2 + L„, which would give us back the .03937 itself. 

tot 

T f T =. 03937007, which is only ^^^^ larger 
.03937. It is not often that an approximation will 
come so near as this. 

This ratio, 5 to 127, is used in cutting millimeter Practical use 

of the foregoing 

thread screws. If the leading screw of the lathe isExampie. 
1 to one inch, the change gears will have the ratio of 
5 to 127; if 8 to one inch, the ratio will be 8 times 
as large, or 40 to 127; so that with leading screw 8 to 
inch, and change gears 40 and 127, we can cut milli- 
meter threads near enough for practical purposes. 

The foregoing operations are more tedious in de- 
scription than in use. The steps have been carefully 
noted, so that the reason for each step can be seen 
from rules of common arithmetic, the operations being 
merely reducing complex fractions. The reductions, 
■gig-, T 2 T , T ^j, etc., are called conver gents, because they 
come nearer and nearer to the required .03937. The 
operations can be shortened as follows: 

Let us find the fractions converging towards .7854", Example. 
the circular pitch of 4 diametral pitch, .7854= T \ 8 Tr 5 / (r ; 
reducing to lowest terms, we have -§-§-§-£. Applying 
the operation for the greatest common divisor: 

3927) 5000 (1 
3927 

1073) 3927 (3 
3219 

708) 1073 (1 
_708 
365) 708 (1 
365 

843) 365 (1 
343 

22) 343 (15 
22 

123 
110 

13) 22 (1 
13 

9) 13 (1 
9 

4) 9 (2 

8 

1) 4 (4 
4 



Bringing the various incomplete quotients as de- 
nominators in a continued fraction as before, we have : 



136 BROWN & SHARPE MFG. CO. 

1 + 1 



3 + 1 

1 + 1 



1 + 1 _ 

1 + 1 _ 
15 + 1 

1 + 1_ 

i + i ; 

2 + ^ 

Now arrange each partial quotient in a line, thus : 
13111 15 1 1 2 4 

1 3. 4 1 J_l ±T_2 183 3 5 5 8_93_ 3 9 27 

x 4 1 9" 14 219 233 4~T2 T13 1 "5"0 

Now place under the first incomplete quotient the 
first reduction or convergent -f, which, of course, is 1 ; 
put under the next partial quotient the next reduction or 

convergent \— 7 or ^ which becomes f . 

1 is larger than .7854, and J is less than .7854. 
Having made two reductions, as previously shown, 
we can shorten the operations by the following rule for next 
eonvergents : Multiply the numerator of the convergent 
just found by the denominator of the next term of the con- 
tinued fraction, or the next incomplete quotient, and add 
to the product the numerator of the preceding convergent ; 
the sum will be the numerator of the next convergent. 
Proceed in the same way for the denominator, that 
is multiply the denominator of the convergent just 
found by the next incomplete quotient and add to the 
product the denominator of the preceding convergent ; 
the sum will be the denominator of the next convergent. 
Continue until the last convergent is the original frac- 
tion. Under each incomplete quotient or denominator 
from the continued fraction arranged in line, will be 
seen the corresponding convergent or reduction. The 
convergent \\ is the one commonly used in cutting 
racks 4 P. This is the same as calling the circumference of 
a circle 22-7 when the diameter is one (1) ; this is also the 
common ratio for cutting any rack. The equivalent decimal 
to \\ is .7S57 X , being about x g large. In three set- 
tings for rack teeth, this error would amount to about .001" 
For a worm, this corresponds to -if- threads to 1"; 
now, with a leading screw of lathe 3 to 1", we would 
want gears on the spindle and screw in a ratio of 33 
to 14. 

Hence, a gear on the spindle with 66 teeth, and a 
gear on the 3 thread screw of 28 teeth, would enable 
us to cut a worm to fit a 4 P gear. 



CHAPTER X. 



ANGLE OF PRESSURE. 



137 



In Fig. 69, let A be any flat disk lying upon a hori- 
zontal plane. Take any piece, B, with a square end, 
a b. Press against A with the piece B in the direction 
of the arrow. 




Fig. 69. 




Fig. 



It is evident A will tend to move directly ahead of 
B in the normal line c d. Now (Fig. 70) let the piece 
B, at one corner f, touch the piece A. Move the piece 
B along the line d e, in the direction of the arrow. 

It is evident that A will not now tend to move in 
the line d e, but will tend to move in the direction of 
the normal c d. When one piece, not attached, presses 
against another, the tendency to move the second 
piece is in the direction of the normal, at the point of 
contact. This normal is called the line of pressure. ^ineofPreas- 

J L ure. 

The angle that this line makes with the path of the 
impelling piece, is called the angle of pressure. 

In Part I., Chapter IV., the lines B A and B A' are 
called lines of pressure. This means that if the gear 



138 



BE0WX k SHAEPE MFG. CO. 



drives the rack, the tendency to move the rack is not 
in the direction of pitch line of rack, but either in the 
direction B A or B A', as we turn the wheel to the left 
or to the right. 

The same law holds if the rack is moved in the 
direction of the pitch line; the tendency to move the 
wheel is not directly tangent to the pitch circle, as if 
driven by a belt, but in the direction of the line of 
pressure. Of course the rack and wheel do move in 
the paths prescribed by then connections with the 
framework, the wheel turning about its axis and the 
rack moving along its ways. This pressure, not in a 
direct path of the moving piece, causes extra friction 
in all toothed gearing that cannot well be avoided. 

Although this pressure works out by the diagram, 
as we have shown, yet, in the actual gears, it is not at 
all certain that they will follow the law as stated, 
because of the friction of teeth among themselves. If 
the driver in a train of gears has no bearing upon its 
tooth-flank, we apprehend there will be but little 
tendency to press the shafts apart. 

The arc through which a wheel passes while one of 
its teeth is in contact is called the arc of action. 
Base of Sys- Until within a few vears, the base of a svstem of 

tern of Inter- J ' 

change a bie c | iible-cmve interchangeable gears was 12 teeth. It 

Gears. o fc> 

is now 15 teeth in the best practice (see Chapter VII., 
Part I.) 

The reason for this change was : the base, 15 teeth, 
gives less angle of pressure and longer arc of contact, 
and hence longer lifetime of gears. 



Arc of Action. 



139 



CHAPTER XL 
INTERNAL GEARS. 



In Part L, Chapter VIII., it is stated that the space 
of an internal gear is the same as the tooth of a spur 
gear. This applies to involute or single-curve gears as 
well os to double-curve gears. 

The sides of teeth in involute internal gears are 
hollowing. It, however, has been customary to cut 
internal gears with spur gear-cutters, a No. 1 cutter 
generally being used. This makes the teeth sides 
convex. Special cutters should be made for coarse Special Cut, 

. , , , , -, t -i . . . , , ters for coarse 

pitch double-curve gears. In designing internal gears, Pitch. 
it is sometimes necessary to depart from the system 
with 15-tooth base, so as to have the pinion differ from 
the wheel by less than 15 teeth. The rules given in 
Part I., Chapters VII. and VIII., will apply in making 
gears on any base besides 15 teeth. If the base is 
low-numbered and the pinion is small, it may be neces- 
sary to resort to the method given at the end of Chap- 
ter VII., because the teeth may be too much rounded 
at the points by following the approximate rules. 
The base must be as small as the difference between Base f or in- 

t>6ril£Ll Ct P ft t* 

the internal gear and its pinion. The base can be Teeth, 
smaller if desired. 

Let it be required to make an internal gear, and 
pinion 24 and 18 teeth, 3 P. Here the base cannot 
be more than 6 teeth. 

In Fig. 71 the base is 6 teeth. The arcs A K and 
O k, drawn about T, have a radius equal to the radius 
of the pitch circle of a 6-tooth gear, 3 P, instead of a 
15-tooth gear, as in Chapter VIII., Part I. 

The outline of teeth of both gears and pinion is pescription of 
made similar to the gear in Chapter VIII. The same 



140 



BROWN & SHARPE MFG. CO. 




GEAR, 24 TEETH. 
PINiON, 18 TEETH, 3P 

P = 3 

N =24 and 18 
P'= 1.0472" 
t— 5236" 
S= .3333" 
D"= .6666" 
}+/= -3857" 
P"+/= 7190" 



NTERNAL GEAR AND PINION IN MESH 



PROVIDENCE, R. I. 141 

letters refer to similar parts. The clearance circle is, 
however, drawn on the outside for the internal gear. 
As before stated, the spaces of a spur wheel become 
the teeth of an internal wheel. The teeth of internal 
gears require but little for fillets at the roots ; they 
are generally strong enough without fillets. The 
teeth of the pinion are also similar to the gear in 
Chapter VIII., substituting 6-tooth for 15-tooth base. 
To avoid confusion, it is well to make a complete 
sketch of one gear before making the other. The arc 
of action is longer in internal gears than in external 
gears. This property sometimes makes it necessary 
to give less fillets than in external gears. 

In Fig. 71 the angle K T A is 30° instead of 12°, as 
in Fig. 12. This brings the line of pressure L P at 
an angle of 60° with the radius C T, instead of 78°. 
A system of spur gears could be made upon this 
6-tooth base. These gears would interchange, but no 
gear of this 6-tooth system would mesh with a double- 
curve gear made upon the 15-tooth system in Part 1. 






c:-:.-.?7z?. 






:• :-Eii3': 



_ . . . _- _ 



'" -. _ : : '. : - _ _ _ : ".. '. -. : : \-\ r-_ : : ; ; : 

:k. 117 lri_ :-r : _'.t ; _ 
:5r ;i-r:, -1 :: • sZ- 



■ - z -■■- 1 


-. — ^ i - - - - ■ - 


.»__ . . . _ 


: _ -• : . : 


: : ; :: _ :: 




- . . . . - . . 






_ :-55~ 


7 T i 7 _U.~ ^ JT 


= :: ~; 


. . . . — . T . 


Iirjr _' - : : - :.'. r :: 


;iLS 4 "* ~ 


-• ~^~- 






7Z T 7Z 








:: -tt. 


- i.— 




-'-^ 




- 


. 




— • 


: 


- 


1 1-4 


~ - 


40 


1460 


€ 


1 9-16 


' - 


-' 


2220 


5 


--- 




; - 


24* 


— . . -,.._-_, 


. -\ 1 ;:r£ 


1 : : t _ -. 




— - ; ~-n . 








— — : V^r 1 szir 


rrS5— r i: 1- 


3 of the 


: :rr^o"_i* 






- ~ -. . li ■ T ... 







10 Fitdb 353 1-3 Ubs. at the Pitch line. 
8 - _ 

6 "40 

823 1-3 

hen it is 2} times the cJrealar pitch. 



PROVIDENCE. II. I. 



143 



TABLE OF DECIMAL EQUIVALENTS 



of 8ths, 16ths, 32nds and 64ths of an inch. 



8ths. 

i=.126 

i=.250 

#=.375 
i = .500 

f=.750 
£=.875 

16ths. 

T V=.0625 
A=-1875 
f V=3125 

T V=.4375 
T 9 g = .5625 
U = .6875 
|f =.8125 
if =.9375 

32nds. 

J^ = .03125 
A=. 09375 

A=. 15625 

j - 

32" 



21875 



.28125 
.34375 
.40625 
.46875 
.53125 
.59375 
.65625 
.71875 
.78125 
.84375 
.90625 
.96875 

64ths. 

^ = . 015625 
A=- 046875 

5 — 
6 4 — 

7 — 
64 — 
_9_ — 
64 — 
11 — 
64 — 
13 — 
64 — 
15. — 
64 — 
±1 — 
64 — 



9_ 
32 
11 
32 

1 Ji 
32 
15. 
32 
LI 
32 
19- 
"3 2 

2 1_ 
32 
2.3. 
32 
25 
32 
21- 

3 2 
_2_iL 
32 
ill- 
32 



.078125 
.109375 
.140625 
.171875 
.203125 
.234375 
.265625 



JL.9 

64 

2.1 

64 

2.3. 

64 

2.5. 

64 

2JT 

64 

29 

64 

_3_1 

64 

3.3. 

64 

3.5. 

"6 4 

3.7 

64 

3.9. 

64 

41 

64 

43 

64 

15 

64 

11 

64 

19. 

64 

5_J_ 

64' 

5.3 

64" 

A5_- 

64' 

£1- 

64" 

II- 

64" 

6.1- 

64" 

6.3- 

64" 



. 296875 
.328125 
.359375 
.390625 
.421875 
.453125 
. 484375 
.515625 
. 546875 
.578125 
.609375 
. 640625 
.671875 
. 703125 
734375 
. 765625 
. 796875 
.828125 
. 859375 
: 890625 
. 921875 
.953125 
.984375 



144 



BBOWH fc SHABPE MFG. CO. 



GEAB WHEEL-. 



TABLE OF TOOTH PAP.TS nBCUTAP. PIT 3H IX TTRST C 1 LUiiy. 



- 
- 



- 



= :=- 



— 



n - — r 



7 



.- — 
- 



P 






: 



:- f YY'+f p .31 p ■-: r 



o 


3 
- 


1.5708 


i . o :• :■•:■ 


0360 


1.2732 


.7360 


1.3732 


:_: 6700 


1 = 


? 


1.6755 


.9375 


.596- 


L.1937 


. ' : "90'i 


1.2874 


.5813 .6281 


1-S- 


4 


1.7952 


.8750 


. 5570 


1.1141 


. 6445 


1.201c 


.5425 .3-:: 


If 


8 

13 


1.9333 


.$125 


.5173 


1 . 0345 


.5985 


1.1158 


5038 .5444 


1: 


2 
3 


- , D944 


.7500 


.4775 


.9549 


. 5525 


1.0200 


.4650 .5025 


1-"- 

± 1G 


16 
23 


2.1855 


.7187 


.4576 


. 9151 


.5201 


.9870 


.4456 .4816 


11 


8 
11 


2.2848 


875 


.4377 


-754 


.5064 


.9441 


.4262 .41 


1 'I 


16 
»1 


2.31 


.6562 


.4178 


.8 


.4834 


0- 


4069 4397 

1 


n 


4 
a 


2.5133 


.6252 


3979 


.7958 


.4604 


.8583 


3875 .4188 


1A 


19 


2.645C 


.5937 


.37-: 


.7" ; : 


.4374 


.8156 


.3681 .3978 


H 


9 


2.7925 


.5625 


.35-1 


.7162 


.4143 


.7724 


.3488 37G9 


1-- 

A 16 


J_6 


2.9568 


.5312 


.3382 


.6764 


.3. : '1; 


.7295 


.3294 .3559 


1 


1 


3.1416 


. 5000 


3183 


.■ 


.3683 


—:■: 


- : : 335C 


15 
16 


ItV 


3.3510 


.4687 


.2984 


. 5968 


.3453 


.6437 


_ : 6 . 3141 


7 
8 


H 


3.5904 


.4375 


.2785 


.557 


.3223 


.6007 


.2713 .2931 


1 3 
16 


1* 


3.866C 


.4: 52 


.2586 


.5173 


.2993 


.5579 


.2519 .2722 


3. 


i- 


4.1858 


.3750 


.2387 


.4775 


.2762 


. 5 1 5 _ 


.2325 .2513 


1 1 
16 


x ll 


4.5 


.3437 


.21 SO 


.4377 


2532 


.4"2: 


.2131 .2303 


2 

3 


u 


4.7124 


.3333 


. 2122 


.4244 


.2455 


.4577 


. .2233 



PROVIDENCE, R. I. 



Ho 



TABLE OF TOOTH PARTS.— Continued. 



CIRCULAR PITCH IN FIRST COLUMN. 



e3 ■ 

11 

P' 

5 

8 


Threads or 

Teeth' per inch 

Linear. 


Diametral 
Pitch. 


Thickness of 

Tooth on 

Pitch Line. 


Addendum 
and ~- 


ft 

hflO 
.g E-" 

G C 

D" 


Depth of Space 

below 

Pitch Line. 


ft ~ 


Width of 

Thread-Tool 

at End. 


Width of 
Thread at Top. 


1 " 
w 

1 3 


P 


t 


s 


«+/ 


.4291 


Px.31 
.1938 


P'x.S35i 


5.0265 


.3125 


.1989 


.3979 


.2301 


.2094 


9 
1 G 


IT 

L 9 


5.5851 


.2812 


.1790 


.3581 


.2071 


.3862 


.1744 


.1884 


1 

2 


2 


6.2832 


.2500 


.1592 


.3183 


.1842 .3433 


.1550 


.1675 


7 
16 


2f 


7.1808 


.2187 


.1393 


.2785 


.1611 .3003 


.1356 


.1466 


2 

"5 


2i 


7.8540 


.2000 


.1273 


.2546 


.1473 .2746 


.1240 


.1340 


3 

8 


2| 


8.3776 


.1875 


.1194 


.2387 


.1381 


.2575 


.1163 


.1256 


1 
3 


3 


9.4248 


.1666 


.1061 


.2122 


.1228. 2289 


.1033 


.1117 


5 

1 G 


H 


10.0531 


.1562 


.0995 


.1989 


.1151 


.2146 


.0969 


.1047 


2 

1 


3} 


10.9956 


.1429 


.0909 


.1819 


.1052 


.1962 


.0886 


.0957 


1 
4 


4 


12.5664 


.1250 


.0796 


.1591 


.0921 


.1716 


.0775 


.0838 


2 

9~ 


4-L 
*2 


14.1372 


.1111 


.0707 


.1415 


.0818 


.1526 


.0689 


.0744 


1 
"5 


5 


15.7080 


1000 


.0637 


.1273 


.0737 


.1373 


.0620 


.0670 


3 

1 G 


°3 


16.7552 


.0937 


.0597 


.1194 


.0690 


.1287 


.0581 


.0628 


I 


G 


18.8496 


.0833 


.0531 


.1061 


.0614 


.1144 


.0517 


.0558 


1 
1 


7 


21.9911 


0714 


.0455 


0910 


.0526 


.0981 


.0443 


.0479 


1 
8 


8 


25.1327 


.0625 


.0398 


.0796 


.0460 


.0858 


.0388 


.0419 


1 

9 


9 


28.2743 


.0555 


.0354 


.0707 


.0409 


.0763 


.0344 


.0372 


1 

1 


10 


31.4159 


.0500 


.0318 .0637 


.0368 


.0687 


.0310 


.0335 


1 
1 6 


1G 


50.2655 


.0312 


.0199 .0398 


.0230 


.0429 


.0194 


.0209 i 



14 'J 



BROWN & SHABTE ilFG. CO. 



GEAR WHEELS. 



TABLE OF TOOTH PART; 



-DIAMETRAL PITCH EN FIRST COLUMN. 



■a 

— « 




T- Z — 
71 — 


< 






5 C 


p 


P 


t 


s 


D' 


»+/ 


D"+/ 


1 

2 


6.2832 


3.1416 


2 . 0000 


4.0000 


2.3142 


4.3142 


a 

4 


4.1888 


2 . 0944 


1.3333 


2.6666 


1 . 5428 


2.8761 


1 


3.1416 


1 . 5708 


1.0000 


2 . 0000 


1 .1571 


2.1571 


li 


2.5133 


1 , 2566 


. 8000 


1.6000 


.9257 


1.7257 


H 


2 . 0944 


1.0472 


.6666 


1 . 3333 


.7714 


1.4381 


If 


1.7952 


.8976 


.5714 


1 1429 


.6612 


1.2326 


2 


1.5708 


. 7854 


.5000 


1 . 0000 


. 5785 


1 . 0785 


01 


1.3963 


.6981 


.4444 


.8888 


.5143 


. 9587 


2i 


1.2566 


.6283 


.4000 


.8000 


.4628 


.8628 


-4 


1.1424 


.5712 


.3636 


.7273 


.4208 


.7844 


3 


1 . 0472 


. 5236 


. 3333 


.6666 


. 3857 


.7190 


31 


.S976 


.4488 


.2857 


.5714 


.3306 


.6163 


4 


. 7854 


.3927 


.2500 


. 5000 


.2893 


. 5393 


5 


.6283 


.3142 


.2000 


.4000 


.2314 


.4314 


G 


.5236 


.2618 


.1666 


.3333 


.1928 


.3595 


i 


.4488 


.2244 


. 1429 


.2857 


. 1653 


.3081 


8 


.3927 


.1963 


.1250 


. 2500 


.1446 


.2696 


9 


.3491 


. 1745 


.1111 


2222 


.1286 


.2397 


10 


.3142 


.1571 


.1000 


.2000 


.1157 


.2157 


11 


.2856 


. 142S 


. 0909 


.1818 


.1052 


.1961 


12 


.2618 


. 1309 


0S33 


.1666 


. 0964 


.1798 


13 


.2417 


. 1208 


.0769 


.1538 


.0890 


.1659 


14 


.2244 


.1122 


.0714 


. 1429 


. 0S26 


.1541 



PROVIDENCE. R. I. 



117 



TABLE OF TOOTH PARTS— Continued. 



DIAMETRAL PITCH IN FIRST COLUMN. 



Diametral 
Pitch. 


Circular 
Pitch. 


Thickness 
of Tooth on 
Pitch Line. 


Addendum 
and J' 


Working Depth 
of Tooth. 


Depth of Space 

below 

Pitch Line. 


Whole Depth 
of Tooth. 


P. 


P'. 


t. 


s. 


D". 


.0771 


D"+/. 


15 


.2094 


.1047 


.0666 


.1333 


.1438 


16 


.1963 


.0982 


.0625 


.1250 


.0723 


.1348 


17 


.1848 


.0924 


.05 8 


.1176 


.0681 


.1269 


18 


.1745 


.0873 


.0555 


.1111 


.0643 


.1198 


19 


.1653 


.0827 


.0526 


.1053 


.0609 


.1135 


20 


.1571 


.0785 


.0500 


.1000 


.0579 


.1079 


22 


.1428 


.0714 


.0455 


.0909 


.0526 


.0980 


24 


.1309 


.0654 


.0417 


.0833 


.0482 


.0898 


26 


.1208 


.0604 


.0385 


.0769 


.0445 


.0829 


28 


.1122 


. 0561 


.0357 


.0714 


.0413 


.0770 


30 


.1047 


.0524 


.0333 


.0666 


.0386 


.0719 


32 


.0982 


.0491 


.0312 


.0625 


.0362 


.0674 


34 


.0924 


.0462 


.0294 


.0588 


.0340 


.0634 


36 


.0873 


.0436 


.0278 


.0555 


.0321 


.0599 


38 


.0827 


.0413 


.0263 


.0526 


.0304 


.0568 


40 


.0785 


.0393 


.0250 


. 0500 


.0289 


.0539 


42 


.0748 


.0374 


.0238 


.0476 


.0275 


.0514 


44 


.0714 


.0357 


.0227 


.0455 


.0263 


.0490 


46 


.0683 


.0341 


.0217 


.0435 


.0252 


.0469 


48 


.0654 


.0327 


.0208 


.0417 


.0241 


.0449 


50 


.0628 


.0314 


.0200 


.0400 


.0231 


.0431 


56 


.0561 


.0280 


.0178 


.0357 


.0207 


.0385 


60 


.0524 


.0262 


.0166 


.0333 


.0193 


. 0360 



us 



BROWN & SHARPE MFG. CO. 



NATURAL SINE. 



T>eg. 


0' 


10' 


20' 


33' 


40' 


50' 


60' 


: 





.00000 


.00291 


.00581 


.00872 


.01163 


.01454 


.01745 


89 


1 


.01745 


.02036 


.02326 


.02617 


.02908 


.03199 


.03489 


: 88 


2 


.03489 


.03780 


. 04071 


.04361 


.04652 


.04943 


.05233 


i 87 


3 


.052S3 


.05524 


.05814 


.06104 


.06395 


.06685 


.06975 


1 80 


4 


.06975 


.07265 


.07555 


.07845 


.08135 


.08425 


.08715 


85 


5 


.08715 


.09005 


.09295 


.09584 


.09874 


.10163 


. 10452 


j 84 


6 


.10452 


.10742 


.11031 


.11320 


.11609 


.11898 


.12186 


, 83 


7 


.12180 


.12475 


.12764 


.13052 


.13341 


. 13629 


.13917 


i 82 


8 


.13917 


.14205 


.14493 


. 14780 


.15068 


.15356 


.15643 


' 81 


9 


.15643 


.15980 


.16217 


.16504 


.16791 


.17078 


.17364 


i 80 


10 


. 17364 


.17651 


.17937 


.13223 


.18509 


.18795 


.19080 


■ 79 


11 


.19080 


.19366 


.19651 


.19936 


.20221 


.20506 


.20791 


78 


12 


.20791 


.21075 


.21359 


.21644 


.21927 


.22211 


.22495 


77 


13 


.22495 


.22778 


.23061 


.23344 


.23627 


.23909 


.24192 


! 7G 


14 


.24192 


.24474 


.24756 


.25038 


.25319 


.25600 


.25881 


75 


15 


.25881 


.26162 


.26443 


.26723 


.27004 


.27284 


.27563 


■ 74 


16 


.27563 


.27843 


.28122 


.28401 


.28680 


.28958 


.29237 


73 


17 


.29237 


.29515 


.29793 


.30070 


.30347 


.30624 


.30901 


1 72 


18 


.30901 


.31178 


.31454 


.31733 


.32000 


.32281 


.32556 


71 


19 


.32556 


.3283L 


.3310G 


.33380 


.33654 


.83928 


. 34202 


70 


20 


.34202 


.34475 


.34748 


.35020 


.35293 


.35565 


.35836 


69 


21 


.35836 


.36103 


.36379 


.36650 


.36920 


.37190 


.37460 


68 


22 


.37460 


.37780 


.37999 


.38268 


.38536 


.38805 


.39073 


! 67 


23 


.39073 


.39340 


.39607 


.39874 


.40141 


.40407 


.40673 


; 66 


24 


.40673 


.40989 


.41204 


.41469 


.41733 


.41998 


.42261 


65 


25 


.42261 


.42525 


.42788 


.43051 


.43313 


.43575 


.43887 


64 


26 


.43837 


.44098 


.44359 


.44619 


.44879 


.45139 


.45399 


63 


27 


.45399 


.45658 


.45916 


.46174 


.46482 


.46690 


.46947 


62 


28 


.46947 


.47203 


.47460 


.47715 


.47971 


.48226 


.48481 


61 


29 


.48481 


.48735 


.48989 


.49242 


.49495 


.49747 


.50000 


60 


30 


.50000 


.50251 


.50503 


.50753 


.51004 


.51254 


.51503 


59 


31 


.51503 


. 51752 • 


.52001 


.52249 


.52497 


.52745 


.52991 


58 


32 


.52991 


.53288 


.53484 


.53730 


.53975 


.54219 


.54463 


57 


33 


.54463 


.54707 


.54950 


.55193 


.55436 


.55677 


.55919 


56 


34 


.55919 


.56160 


.56400 


.56640 


.56880 


.57119 


.57357 


55 


35 


.57357 


. 57595 


.57833 


. 58070 


.58306 


.58542 


.58778 


54 


36 


.58778 


.59013 


.59243 . 


.59482 


.59715 


.59948 


.60181 


53 


37 


.60181 


.60413 


.60645 


.60876 


.61106 


.61336 


.61566 


52 


38 


.61566 


.61795 


.62023 


.62251 


.62478 


.62705 


.62932 


51 


39 


.62932 


.63157 


.63383 


.63607 


.63832 


.64055 


.64278 


50 


40 


.64278 


.64501 


.64723 


.64944 


.65165 


.65386 


.65605 


49 


41 


.65605 


.65825 


.66043 


.66262 


. 66479 


.66696 


.66913 


48 


42 


.66913 


.67128 


.67344 


.67559 


.67773 


.67986 


.68199 


47 


43 


.68199 


.68412 


.68624 


. 68835 


.69046 


.69256 


.69465 


46 


44 


.69465 


.69674 


.69883 


.70093 


.70298 


. 70504 


. 70710 


45 


■ 


60' 


50' 


40' 


30' 


2Y 


10' 


C 

i 


~Deg. 



NATURAL COSINE. 



PROVIDENCE. R. I. 



149 



NATURAL SINE. 



Beg. 


0' 


! 10 ' 


20' 


30' 


40' 


50' 


60' 


j 


45 


.70710 


| .70916 


.71120 


.71325 


. 71528 


.71731 


.71934 


44 


46 


.71934 


. 72185 


.72336 


.72537 


. 72737 


. 72936 


.73135 


43 


47 


. 73185 


. 73333 


. 73530 


.73727 


. 73923 


.74119 


.74314 


42 


48 


74314 


.74508 


.74702 


.74895 


.75088 


.75279 


.75471 


41 


49 


.75471 


.75661 


. 75851 


. 76040 


.76229 


.76417 


.76604 


40 


50 


.76U04 


.76791 


.76977 


.77162 


.77347 


.77531 


.77714 


39 


51 


.77714 


.77897 


.78079 


.78260 


. 78441 


.78621 


. 78801 


38 


53 


.78801 


.78979 


.79157 


.79385 


.79512 


.79688 


. 79863 


37 


53 


.79863 


.80038 


.80212 


.80385 


.80558 


.80730 


.80901 


36 


54 


.80901 


.81072 


.81242 


.81411 


.81580 


.81748 


.81915 


iiO 


55 


.81915 


.82081 


.82247 


.82412 


.82577 


.82740 


.82903 


34 


56 


.82903 


.83066 


.83227 


.83383 


.83548 


.83708 


.83867 


33 


57 


.83867 


.84025 


.84182 


.84339 


.84495 


. 84650 


.84804 


32 


53 


.84804 


.84958 


.85111 


.85264 


.85415 


.85566 


.85716 


31 


59 


.85716 


.85866 


.86014 


.86162 


.88310 


.86456 


. 86602 


30 


GO 


.86602 


.86747 


.86892 


.87035 


.87178 


.87320 


.87462 


29 


Gl 


.87462 


.87602 


. 87742 


.87881 


.88020 


.88157 


.88294 


28 


G2 


.88294 


.88430 


.88566 


.88701 


. 88835 


.88968 


.89100 


27 


03 


.89100 


.89232 


.89363 


.89493 


.89622 


.89751 


.89879 


£6 


64 


.89879 


.90006 


.90132 


.90258 


.90383 


.90507 


. 90630 


25 


65 


.90630 


.90753 


.90875 


.90996 


.91116 


.91235 


.91354 


24 


G6 


.91854 


.91472 


.91589 


.91706 


.91821 


.91936 


.92050 


23 


67 


.92050 


.92163 


.92276 


.92388 


.92498 


.92609 


.92718 


22 


G8 


.92718 


. 92827 


.92934 


.93041 


.93148 


.93253 


. 93358 


21 


G9 


.93358 


.93461 


.93565 


.93667 


.93768 


.93869 


. 93969 


20 


70 


.93969 


.94068 


.94166 


.94264 


.94360 


.94456 


.94551 


19 


71 


.94551 


.94646 


.94789 


.94832 


.94924 


.95015 


.95105 


18 


72 


.95105 


.95195 


.95283 


.95371 


* .95458 


.95545 


.95630 


17 


73 


.95630 


.95715 


.95799 


.95882 


.95964 


.96045 


.98126 


16 


74 


.96126 


.96205 


.96284 


. 96363 


.96440 


.96516 


. 98592 


15 


75 


.96592 


.96667 


.96741 


.96814 


.96887 


.96958 


.97029 


14 


76 


.97029 


.97099 


.97168 


.97237 


.97304 


.97371 


.97437 


13 


77 


.97437 


.97502 


.97566 


.97623 


.97692 


.97753 


.97814 


12 


78 


.97814 


.97874 


.97934 


.97992 


.98050 


.98106 


.98162 


11 


79 


.98162 


.98217 


.98272 


.98325 


.98378 


.98429 


.98480 


10 


80 


.98480 


.98530 


.98580 


.98628 


.98676 


. 98722 


.98768 


9 


81 


.98768 


.98813 


.98858 


.98901 


.98944 


.98985 


.99026 


8 


82 


. 99026 


.99066 


.99106 


.99144 


.99182 


.99218 


.99254 


7 


83 


.99254 


.99289 


.99323 


.99357 


.99889 


.99421 


.99452 


6 


84 


.99452 


.99482 


.99511 


.99539 


.99567 


.99593 


.99619 


5 


85 


.99619 


.99644 


.99668 


.99691 


.99714 


. 99735 


.99756 


4 


86 


.99756 


.99776 


.99795 


99813 


.99880 


.99847 


.99863 


3 


87 


.99863 


.99877 


.99891 


.99904 


.99917 


.99928 


. 99939 


2 


88 


.99939 


.99948 


.99957 


.99965 


.99972 


.99979 


.99984 


1 


89 


.99984 


.99989 


.99993 


.99993 


.99998 


.99999 


1.0000 





"" 


60' 


50' 


40' 


30' 


£0' 


w 


0' 


Deg. 



NATURAL COSINE. 



150 



BUOWN & SHARPK MFG. CO. 



NATURAL TANGENT. 



Dej;. ; 


0' 


10' 


20' 


80' 


40' 


50' 


60' 


———— 





.00000 


.00290 


.00581 


. 00872 


.01168 


.01454 


.01745 


89 


1 


.01745 


.02036 


.02327 


.02618 


.02909 


.03200 


.03492 


88 


2 


.03492 


.03783 


.04074 


.04366 


.04657 


.04949 


.05240 


87 


3 


.05240 


.05532 


.05824 


.06116 


.06408 


.06700 


.06992 


86 


4 


.06992 


.07285 


.07577 


.07870 


.08162 


.08455 


.08748 


85 


5 


.08748 


.09042 


.09335 


.09628 


.09922 


.10216 


.10510 


84 


(5 


.10510 


.10804 


.11099 


.11393 


.11688 


.11983 


. 12278 


83 


7 


.12278 


.12573 


.12869 


.13165 


.13461 


. 13757 


.14054 


82 


8 


.14054 


.14350 


.14647 


. 14945 


.15242 


.15540 


.15838 


81 


9 


.15838 


.16136 


. 16435 


. 16734 


.17033 


.17332 


.17632 


80 


10 


.17632 


. 17932 


. 18233 


.18533 


.18834 


.19186 


.19438 


79 


11 


. 19438 


. 19740 


.20042 


.20345 


.20648 


.20951 


.21255 


78 


12 


.21255 


.21559 


.21864 


.22169 


.22474 


.22780 


.23086 


77 


13 


.23086 


.23393 


.23700 


.24007 


.24315 


.24624 


.24932 


76 


14 


.24932 


.25242 


.25551 


.25861 


.26172 


.26483 


.26794 


75 


15 


.26794 


.27106 


.27419 


.27732 


.28046 


.28360 


.28674 


74 


16 


.28674 


. 28989 


.29305 


.29621 


.29938 


.30255 


.30578 


73 


17 


.30573 


.30891 


.31210 


.31529 


.31850 


.32170 


.32492 


72 


18 


.32492 


.32813 


.33136 


.33459 


.33783 


.34107 


.34432 


71 


19 


.34482 


. 34758 


.35084 


.35411 


.85789 


.36067 


.36397 


70 


20 


.36397 


.3672(5 


.37057 


.37388 


.37720 


. 38053 


.38386 


69 


21 


.38386 


.38720 


.39055 


.39391 


.39727 


.40064 


.40402 


68 


22 


.40402 


.40741 


.41080 


.41421 


.41762 


.42104 


.42447 


67 


23 


.42447 


.42791 


.43135 


.43481 


.43827 


.44174 


.44522 


60 


24 


.44522 


.44871 


.45221 


. 45572 


.45924 


.46277 


.46630 


65 


25 


.46630 


. 46985 


.47341 


.47697 


.48055 


.48413 


.48773 


64 


26 


.48773 


.49133 


.49495 


.49858 


.50221 


.50586 


.50952 


63 


27 


.50952 


.51319 


.51687 


. 52056 


.52427 


. 52798 


.53170 


62 


28 


.53170 


.53544 


.53919 


. 54295 


. 54672 


.55051 


.55430 


61 


29 


.55480 


.55811 


.56195 


.56577 


.56961 


. 57847 


.57735 


60 


30 


.57785 


.58123 


.58513 


.58904 


.59297 


. 59690 


. 60086 


59 


31 


.60086 


.60482 


.60880 


.61280 


.61680 


.62083 


. 62486 


58 


32 


. 62486 


.62892 


.63298 


.63707 


.64116 


.64528 


.64940 


57 


33 


.64940 


.65355 


.65771 


.66188 


.66607 


.67028 


.67450 


5,5 


84 


.67450 


.67874 


.68300 


.68728 


.69157 


. 69588 


.70020 


55 


35 


.70020 


.70455 


.70891 


. 71329 


.71769 


.72210 


.72654 


5-1 


36 


.72654 


.73099 


.73546 


.73996 


.74447 


.74900 


.75355 


55 


37 


.75355 


.75812 


.76271 


. 76732 


. 77195 


. 77661 


.78128 


52 


38 


.78128 


.78598 


.79069 


.79543 


.80019 


.80497 


. 80978 


51 


31) 


.80978 


.81461 


.81946 


. 82433 


.82923 


.83415 


.83910 


50 


40 


.83910 


.84406 


.84906 


. 85408 


.85912 


.86419 


. 86928 


40 


41 


.86928 


.87440 


. 87955 


.88472 


. 88992 


.89515 


.90040 


48 


42 


.90040 


.90568 


.91099 


.91633 


.921(59 


.92709 


.93251 


47 


4:5 


.93251 


.93796 


.94845 


.94896 


.95450 


.96008 


.96568 


46 


44 


.96568 


! .97132 


.97699 


.982(59 


.118843 


.99419 


1.0000 


45 


* 


CO' 


50' 


40' 


30' 


20' 


10' 


0' 


Deg. 



NATURAL COTANGENT. 



PROVIDENCE, R. I. 



151 



NATURAL TANGENT. 



Beg. 


o' 


10' 


20' 


30' 


40' 


50' 


60 




45 


1.0000 


1.0058 


1.0117 


1.0176 


1.0235 


1.0295 


1.0355 


44 


46 


1.0355 


1.0415 


1.0476 


1.0537 


1.0599 


1.0661 


1.0723 


43 


47 


1.0723 


1.0786 


1.0849 


1.0913 


1.0977 


1.1041 


1.1106 


42 


48 


1.1106 


i.-;m 


1 . 1236 


1 . 1302 


1.1369 


1.1436 


1.1503 


41 


49 


1.1503 


1.1571 


1 1639 


1.1708 


1.1777 


1.1847 


1.1917 


40 


50 


1.1917 


1.1988 


1.2059 


1.2131 


1.2203 


1.2275 


1 2349 


39 


51 


1.2349 


1.2422 


1.2496 


1.2571 


1.2647 


1.2723 


1.2799 


38 


52 


1.2799 


1.2876 


1.2954 


1.3032 


1.3111 


1.3190 


1.3270 


37 


53 


1.3270 


1.3351 


1.3432 


1.3514 


1.3596 


1.3680 


1.3763 


36 


54 


1.3763 


1.3848 


1.3933 


1.4019 


1.4106 


1.4193 


1.4281 


35 


55 


1.4281 


1.4370 


1.4459 


1.4550 


1.4641 


1.4733 


1.4825 


34 


56 


1 4825 


1.4919 


1.5013 


1.5108 


1.5204 


1.5301 


1.5398 


33 


57 


1.5398 


1.5497 


1.5596 


1.5696 


1.5798 


1.5900 


1.6003 


32 


58 


1.6003 


1.6107 


1.6212 


1.6318 


1.6425 


1.6533 


1.6642 


31 


59 


1.6642 


1.6753 


1.6864 


1.6976 


1 . 7090 


1.7204 


1.7320 


30 


60 


1.7320 


1.7437 


1.7555 


1 . 7674 


1.7795 


1.7917 


1.8040 


29 


61 


1.8040 


1.8164 


1.8290 


1.8417 


1.8546 


1.8676 


1.8807 


28 


62 


1.8807 


.1.8940 


1.9074 


1.9209 


1.9347 


1.9485 


1.9626 


27 


63 


1.9626 


1.9768 


1.9911 


2.0056 


2.0203 


2.0352 


2.0503 


26 


64 


2.0503 


2.0655 


2.0809 


2.0965 


2.1123 


2.1283 


2.1445 


25 


65 


2.1445 


2.1609 


2.1774 


2.1943 


2.2113 


2.2285 


2.2460 


24 


66 


2.2460 


2.2637 


2.2816 


2.2998 


2.3182 


2.3369 


2.3558 


23 


67 


2.3558 


2.3750 


2.3944 


2.4142 


2.4342 


2.4545 


2.4750 


22 


68 


2.4750 


2.4959 


2.5171 


2.5386 


2.5604 


2.5826 


2.6050 


21 


69 


2.6050 


2.6279 


2.6510 


2.6746 


2.6985 


2.7228 


2.7474 


20 


70 


2.7474 


2.7725 


2.7980 


2.8239 


2.8502 


2.8770 


2.9042 


19 


71 


2.9042 


2.9318 


2.9600 


2.9886 


3.0178 


3.0474 


3.0776 


18 


72 


3.0776 


3.1084 


3.1397 


3.1715 


3.2040 


3.2371 


3.2708 


17 


73 


3.2708 


3.3052 


3.3402 


3.3759 


3.4123 


3.4495 


3.4874 


16 


74 


3.4874 


3.5260 


3.5655 


3.6058 


3.6470 


3.6890 


3.7320 


15 


75 


3.7320 


3.7759 


3.8208 


3.8607 


3.9136 


3.9616 


4.0107 


14 


76 


4.0107 


4.0610 


4.1125 


4.1653 


4.2193 


4.2747 


4.3314 


13 


77 


4.3B14 


4.3896 


4.4494 


4.5107 


4.5736 


4.6382 


4.7046 


12 


78 


4.7046 


4.7728 


4.8430 


4.9151 


4.9894 


5.0653 


5.1445 


11 


79 


5.1445 


5.2256 


5.3092 


5.3955 


5.4845 


5.5763 


5.6712 


10 


80 


5.6712 


5.7693 


5.8708 


5.9757 


6.0844 


6.1970 


6.3137 


9 


81 


6.3137 


6.4348 


6.5605 


6.6911 


6.8269 


6.9682 


7.1153 


8 


82 


7.1153 


7.2687 


7.4287 


7.5957 


7.7703 


7.9530 


8.1443 


7 


83 


S.1443 


8.3449 


8.5555 


8.7768 


9.0098 


9.2553 


9.5143 


6 


84 


9.5143 


9.7881 


10.078 


10.385 


10.711 


11.059 


11.430 


5 


85 


11.430 


11.826 


12.250 


12.706 


13.196 


13.726 


14.300 


4 


86 


14.300 


14.924 


15.604 


16.349 


17.169 


18.075 


19.081 


3 


87 


19.081 


20.205 


21.470 


22.904 


24.541 


26.431 


28.636 


2 


88 


28.636 


31.241 


34.367 


38.188 


42.964 


49.103 


57.290 


1 


89 


57.290 


68.750 


85.939 


114.58 


171.88 


343 . 77 


00 







60' 


50' 

l 


40' 


30' 


20' 


10' 


0' 


Deg. 



NATURAL COTANGENT. 



152 



BROWN J: SHARPE MFG. CO. 



NATURAL SECANT. 



1 

Beg. 


0' 


10' 


20' 30' 


4/ 


50 


60' 







1.0000 


1 . 0000 


1.0000 


1 . 0000 


1 . 0000 


1.0001 


1 . 0001 


89 


1 


1.000L 


1 . 0002 


1 . 0002 


1.0003 


1.0001 


1 . 0005 


l.OOOo 


88 | 


2 


1 . 0006 


1.0007 


1 0008 


1 . 0009 


1.0010 


1.0012 


1.0013 


87 


3 


1.0013 


1 . 0015 


1.0016 


1.0018 


1.0020 


1 . 0022 


1.0024 


86 i 


4 


1 . 0024 


1 . 002a 


1.0028 


1.0030 


1.0033 


1.0035 


1.0033 


85 


5 


1.0038 


1 . 0040 


1 . 0043 


1.0016 


1.0049 


1.0052 


1 . 0055 


84 | 


6 


; 1.0055 


1 . 0058 


1.0031 


1.0064 


1.0068 


1.0071 


1.0075 


83 


7 


i 1.0075 


1.0078 


1 . 0082 


1 . 0086 


1.0090 


1 . 0094 


1 . 0098 


, 82 


8 


1.0098 


1.0102 


1.0103 


1.0111 


1.0115 


1.0120 


1.0124 


1 81 


9 


1.0124 


1.0129 


1.0134 


1 . 0139 


1.0144 


1 . 0149 


1.0154 


i 80 


10 


1.0154 


1.0159 


1.0164 


1.0170 


1.0175 


1.0181 


1.0187 


; 79 


11 


1.0187 


1 . 0192 


1.0198 


1.0204 


1.0210 


1.02L7 


1 . 0223 


1 78 


12 


1 . 0223 


1.0229 


1 . 02:36 


1.0242 


1 . 0249 


1 . 0256 


1.0263 


77 


u 


1 . 0263 


1.0269 


1.0277 


1 . 0284 


1.0291 


1 . 0298 


1.0303 1 


76 


14 


1.0303 


1.0313 


1.0321 


1 . 0329 


1.0336 


1 . 0344 


1.0352 


75 


15 


1.0352 


1.0380 


1.0369 


1.0377 


1 . 0385 


1 . 0394 


1.0402 


74 


1(3 


1.0402 


1 . 0411 


1 . 0420 


1.0429 


1.0438 


1.0147 


1 . 0456 


73 


17 


1 . 0456 


1.0436 


1 . 0475 


1.0485 


1 . 0494 


1.0504 


1.0514 


1 72 


18 


1.0514 


1.0524 


1 . 0534 


1 . 0544 


1 . 0555 


1.0565 


1.0576 


71 


19 


1.0576 


1.0586 


1.0597 


1.0:08 


1.0319 


1 0630 


1.0641 


70 


20 


1.0641 


1 . 0853 


1.06C4 


1.0.'i76 


i.o::87 


1 . 0699 


1.0711 


C9 


21 


: i.07ii 


1.0723 


1 . 0735 


1 . 0747 


1.0760 


1.0772 


1.0785 


68 


23 


1.0785 


1.0798 


1.0810 


1.0823 


1.0837 


1 . 0850 


1.0833 


67 


23 


1 . 0863 


1.0877 


1 . 0890 


1.0904 


1.0918 


1.0932 


1.0946 


66 


1 24 


1.0946 


1.0960 


1 . 0974 


1.0989 


1.1004 


1.1018 


1.1033 


65 


25 


1 . 1033 


1.1048 


1.1063 


1.1079 


1.1094 


1.1110 


1.1126 


64 


26 


1.1126 


1.1141 


1.1157 


1.1174 


1.1190 


1.1206 


1.1223 


63 


27 


1.1223 


1.1239 


1.1256 


1.1273 


1.1290 


1.1308 


1.1125 


62 


28 


1.1325 


1 . 1343 


1.1361 


1.1378 


1.1396 


1.1415 


1.1433 


61 


29 


1.1433 


1.1452 


1.1470 ' 


1 . 1489 


1.1508 


1 . 1527 


1 . 1547 


60 


, 30 


1.1547 


1 . 1566 


1.1583 


1.1605 


1 . 1625 


1.1646 


1.1665 


59 


1 3L 


1.1666 


1.1686 


1.1707 


1.1723 


1.1749 


1.1770 


1.1791 


58 


32 


1.1791 


1.1813 


1.1835 


1.1856 


1.1878 


1.1901 


1.1923 


57 


33 


1.1923 


1.1946 


1 . 1969 


1.1992 


1.2015 


1.20-8 


1 . 2062 


56 


34 


1.2032 


1.2085 


1.2109 


1.2134 


1.2158 


1.2182 


1.2207 


55 


35 


1.2207 


1.2232 


1.2257 


1.2283 


1.2308 


1.2334 


1.2360 


54 | 


3) 


1.2360 


1.2386 


1.2413 


1.2440 


1 . 24 56 


1.2494 


1.2521 


53 


37 


1.2521 i 1.2548 


1.2576 


1.2304 


1.2632 


1.2661 


1.2690 


52 | 


33 


1.26v 9 1 1.2719 


1.2748 


1.2777 


1.2807 


1.2837 


1.2887 : 


51 


39 


1.2867 


1.2898 


1.2328 


1.2959 


1.2990 


1.3022 


1.3054 


50 


49 


1.3054 


1.3086 


1.3118 


1.3150 


1=3183 


1.3216 


1.3250 


49 


41 


1.32-30 


1.3283 


1.3317 i 


1.3351 : 


1.3336 


1.3421 


1.3456 


48 


42 


1 . 3456 


1.3491 


1.3527 


1.3563 


1.3599 


1.3636 


1.3673 


47 


43 


1 3673 


1.3710 


1.3748 ; 


1.3785 ; 


1.3824 


1.3862 


1.3901 


46 


44 


1.3901 


1.3940 


1.39S0 


1.4020 


1.40J0 1.4101 


1.4142 


45 




1 

GO' 


50' 


10' 


30' 


20' 10' 





Deg. ' 



NATURAL COSECANT. 



PROVIDENCE, R. I. 



153 



NATURAL SECANT. 



1 
Beg. 


0' 


10' 


20' 


30' 


40' 


50' 


60' : 




45 


1.4142 


1.4183 


1.422j 


1.4267 


1.4309 


1.4352 


1.4395 


44 


46 


1.4395 


1.4439 


1.4483 


1.4527 


1.4572 


1.4617 


1.4662 


43 


47 


1.4662 


1.4708 


1.4755 


1.4801 


1.4849 


1.4896 


1.4944 


42 


48 


1.4944 


1.4993 


1 . 5042 


1.5091 


1.5141 


1.5191 


1.5242 


41 


49 


1.5242 


1.5293 


1.5345 


1.5397 


1.5450 


1 . 5503 


1.5557 


40 


50 


1.5557 


1.5611 


1.5666 


1.5721 


1.5777 


1.5833 


1.5890 


39 


51 


1.5890 


1.5947 


1.6005 


1.(3063 


1.6122 


1.6182 


1.6242 


38 


52 


1.6242 


1.6303 


1.6364 


1.6426 


1.6489 


1.6552 


1.6616 


37 


53 


1.6616 


1.6680 


1 . 6745 


1.6811 


1.6878 


1.6945 


1.7013 


36 


54 


1.7013 


1.7081 


1.7150 


1.7220 


1.7291 


1.7362 


1.7434 


| 35 


55 


1.7434 


1.7507 


1.7580 


1.7655 


1.7730 


1.7806 


1.7882 


! 34 


56 


1.7882 


1.7960 


1.8038 


1.8118 


1.8198 


1.8278 


1.8360 


1 33 


57 


1.8360 


1.8443 


1.8527 


1.8611 


1.8697 


1.8783 


1.8870 


1 32 


58 


1.8870 


1.8959 


1.9048 


1.9138 


1.9230 


1.9322 


1.9416 


31 


59 


1.9416 


1.9510 


1.9603 


1.9702 


1.9800 


1.9899 


2.0000 


30 


60 


2.0000 


2.0101 


2.0203 


2 0307 


2.0412 


2.0519 


2.0826 


29 


61 ! 


2.0623 


2.0735 


2.0845 


2.0957 


2.1070 


2.1184 


2.1300 


; 28 


62 ! 


2.1300 


2.1417 


2.1536 


2.1656 


2.1778 


2.1901 


2.2026 


27 


63 


2.2026 


2.2153 


2.2281 


2.2411 


2.2543 


2.2376 


2.2811 


26 


64 


2.2811 


2.2948 


2.3087 


2.3228 


2.3370 


2.3515 


2.3662 


25 


65 


2.3662 


2.3810 


2.3961 


2.4114 


2.4239 


2.4426 


2.4585 


24 


66 


2.4585 


2.4747 


2.4911 


2.5078 


2.5247 


2.5418 


2.5593 


! 23 


67 


2.5593 


2.5769 


2.5949 


2.6131 


2.6316 


2.6503 


2.6694 


22 


68 


2.6694 


2.6883 


2.7085 


2.7285 


2.7488 


2.7694 


2.7904 


'. 21 


69 


2.7904 


2.8117 


2.8334 


2.8554 


2.8778 


2.9006 


2.9238 


20 


70 


2.9238 


2.9473 


2.9713 


2.9957 


3.0205 


3.0458 


3.0715 


19 


71 


3.0715 


3.0977 


3.1243 


3.1515 


3.1791 


3.2073 


3.2360 


18 


i 72 


3.2360 


3.2853 


3.2951 


3.3255 


3.3564 


3.3880 


3.4203 


17 


1 73 


3.4203 


3.4531 


3.4867 


3.5209 


3.5558 


3.5915 


3.6279 


16 


74 


3.6279 


3.6651 


3.7031 


3.7419 


3.7816 


3 8222 


3.8637 


15 


75 


3.8337 


3.9061 


3.9495 


3.9939 


4.0393 


4.0859 


4.1335 


14 


76 


4.1335 


4.1823 


4.2323 


4.2836 


4.3362 


4.3901 


4.4454 


13 


77 


4.4454 


4.5021 


4.5604 


4.6202 


4.6816 


4.7448 


4.8097 


12 


78 


'4.8097 


4.8764 


4.9451 


5 0158 


5.0886 


5.1635 


5.2408 


11 


79 


5.2408 


5.3204 


5.4026 


5.4874 


5.5749 


5.6653 


5.7587 


10 


80 


5.7587 


5.8553 


5.9553 


6.0588 


6.16C0 


6.2771 


6.3924 


9 


81 


6.3924 


6.5120 


6.6363 


6.7654 


6.8997 


7.0396 


7.1852 


8 


82 


7.1852 


7.3371 


7.4957 


7.6612 


7.8344 


8.0156 


8.2055 


17 
i 


83 


8.2055 


8.4046 


8.6137 


8.8336 


9.0651 


9.3091 


9.5667 


6 


84 


9.5667 


9.8391 


10.127 


10.433 


10.758 


11.104 


11.473 


5 


85 


11.473 


11.868 


12.291 


12.745 


13.234 


13.763 


14.335 


4 


86 


14.335 


14.957 


15.633 


16.380 


17.198 


18.102 


19.107 


3 


87 


19.107 


20.230 


21.493 


22.925 


24.562 


26.450 


28 653 


2 


88 


28.653 


31.257 


34.382 


38.201 


42 975 


49.114 


57.298 


1 


89 


57.298 


68.757 


85.945 


114.59 


171.88 


343.77 


00 







GO' 


50 


43' 


80' 


20' 


10' 


0' 


Deg. 



NATURAL COSECANT. 



:.:.4 



SHABPE MFG. 



:o. 



TABLE OF DECIMAL EQIIVALEXTS 



: IHILIMZTEES AXI' FRACTIONS OF MILLrAEETEP.-. 



inm. Inehi s. 



1 . 

5 0" 

_a_. 

•f 

3 - 
5 ' 

50" 
5 
50 

:_ 

« ■ 

' 

' 

9 

50 
1 

' ' 
1 1 

■' ' 

■ : . 
' i 

1 3_- 
* 

Ji 
I - 
1 5 - 
' 
1 6- 

| " - 

LA 
50 

1 S 

S " 
go 

" ' 

'f " 
: - ' 

■: t. 

• r 

|S 

5 



.0007! 

.00157 

236 

.00315 

.00394 

.('0551 

.001 

.007 

.--7-7 

. 00945 
.01024 
.01102 
.01181 
.012 

01339 
.01417 

014 

. 01575 
. -1654 
.017;: 2 
.01811 
.01890 
. 01 ■ 



mm. Inches. 

2047 
02126 

02205 
02283 

02362 
02441 
02520 
02598 

02677 
02756 
02835 
02913 
02992 
03071 
03150 
03228 
03307 
033-6 
03465 
03543 
03622 
03701 
03780 
03858 
03937 



•: : ■ 
a : ' 

Li- 

: 

I s 

■: 
30 

3J_ 



3A 

3 5 

- 6- 

n 

a _ 
I 8.- 

" " 

50 
40- 

" 

" 
1 2- 

50" 
43- 

." 

50" 

. 

S 
ti- 

5 

4 S- 
" 

4 ■ 



1 = 



mrn. 


Inches. 


2 = 


.07874 


3 = 


.11811 


4 = 


.15748 


5= 


.19685 


6= 


.2362- 


i = 


.2755V' 


8 = 


.31496 


,' = 


. 35433 


10= 


.39370 


11 = 


433-7 


12= 


.47244 


13= 


.51181 


14= 


.55118 


15= 


.59055 


16= 


.62992 


17= 


I 


18= 


">•:' 


19= 


.74- 


20= 


.7874C 


21 = 


.-2677 


22= 


34 S14 


23 = 


.90551 


24= 


.94488 


25= 


-425 


26 = 1 


.02:; . 



10 mm. = l Centimeter=0.3937 in<: es 

... =1 Decimeter =c 2 
10 dm. =1 Meter =39.37 

'2~>A mm.=l Enirli>h Inch. 



INDEX. 



A. 

PACK 

Abbreviations of Parts of Teeth and Gears ... 4 

Addendum 2 

Angle, How to Lay OS. an SB, 107 

Angle Increment 106 

Angle of Edge 102 

Angle of Face 104 

Angle of Pressure 137 

Angle of Spiral 113 

Angular Velocity 2 

Annular Gears - 32, 139 

Arc of Action 138 

B. 

• 

Base Circle 11 

Base of Epicycloidal System 25 

Base of Internal Gears 130 

Bevel Gear Blanks 34 

Bevel Gear Cutting on B. & S. Automatic Gear Cutter 52 

Bevel Gear Angles by Diagram 36 

Bevel Gear Angles by Calculation 102, 106 

Bevel Gear, Form of Teeth of 41 

Bevel Gear, Whole Diameter of 36, 104 

C. 

Centers, Line of 2 

Circular Pitch 4 

Classification of Gearing 5 

Clearance at Bottom of Space G 

Clearance in Pattern Gears : 8 

Condition of Constant Velocity Katio 2 

Contact, Arc of '. 138 

Continued Fractions 132 

Coppering Solution 85 

Cutters, How to Order 83 

Cutters, Table of Epicycloidal 8t 



15G INDEX. 

PAGE. 

Cutters, Table of Involute 82 

Cutters, Table of Speeds for 81 

Cutting Bevel Gears on B. & S. Automatic Gear Cutter 52 

Cutting Spiral Gears on a Universal Milling Machine 122 

D. 

Decimal Equivalents, Tables of 143, 154 

Diameter Increment 101 

Diameter of Pitch Circle G 

Diameter Pitch 5 

Diametral Pitch 17 

Distance between Centers 8 

E. 

Elements of Gear Teeth 5 

Epicycloidal Gears, with more and. less than 15 Teeth 30 

Epicycloidal Gears, with 15 Teeth 25 

Epi cycl oid al Rack 27 

F. 

Face, Width of Spur Gear 80 

Flanks of Teeth in Low-numbered. Pinions 20 

G. 

Gear Cutters, How to Order 83 

Gear Patterns ." 8 

Gearing Classified 5 

Gears, Bevel , 31, 41, 102 

Gears, Epicycloidal 25 

Gears, Involute 9 

Gears, Spiral 109, 122 

Gears, Worm G3 

II. 
Herring-bone Gears 130 

I. 

Increment, Angle 100 

Increment, Diameter 104 

Interchangeable Gears 24 

Internal or Annular Gears 32, 139 

Involute Gears, 30 Teeth and over 9 

Involute Gears, with Less than 30 Teeth 20 

Involute Rack 12 



INDEX. 157 

L 

PAGE. 

Lead of a Worm 62 

Limiting Numbers of Teeth in Internal Gears 32 

Line of Centers 2 

Line of Pressure 12, 137 

Linear Velocity 1 

M. 

Machine, B. & S., for Cutting Bevel Gears 52 

N. 

Normal 11G 

Normal Helix 116 

Normal Pitch 116 

0. 
Original Cylinders 1 

P. 

Pattern Gears 8 

Pitch Circle 3 

Pitch, Circular or Linear -4 

Pitch, a Diameter 6 

Pitch, Diametral 17 

Pitch, Normal 116 

Pitch of Spirals 112 

Polygons, Calculations for Diameters of ,. 05 

11 

Rack 12 

Pack for Epicycloidal Gears •. 27 

Pack for Involute Gears 12 

Pack for Spiral Gears 121 

Relative Angular Velocity 2 

Rolling Contact of Pitch Circle 3 

S. 

Screw Gearing 109, 130 

Single-Curve Teeth 9 

Speed of Gear Cutters 81 



158 INDEX. 

PAGE. 

Spiral Gearing 109. 122 

Standard Templets ". 27 

Strength of Gears 142 



Table of Decimal Equivalents 143, 154 

Table of Sines, etc 148, 153 

Table of Speeds for Gear Cutters 81 

Table of Tooth Parts 144, 147 



Velocity, Angular 2 

Velocity, Linear 1 

Velocity. Relative 2 

W. 

Wear of Teeth SO, 129 

Worm Gears 63 



PRACTICAL TREATISE 



ON 



GEARING 



BROWN &- SHARPE MFG. CO. 



Providence, R. I., U. S. A. 




'WT&&* 



/ 



•»* 






-V 






V '>> 






<?> 



O. 









OCT 



V 



> V 



' o 












o. 









A 






- 



*>/ 






\ ^, 










■■' - 









O 





















;** ^ . 



^ 



. 












^ V*. 





















\ -*■ V- yj- 












V 















*> V 






q> ^ 






$ ."fe, 






* 8 « '■ 






*P 


















o 






<h 


















-*v 



cy ' 



c*> 








































. 



